Arturo Giles Flores
email: arturo.giles@cimat.mx

Otoniel N Silva
email: otoniel@im.unam.mx

Bernard Teissier
email: bernard.teissier@imj-prg.fr

Otoniel Nogueira Da Silva

The biLipschitz geometry of complex curves: an algebraic approach

The purpose of these notes is to explain why a generic projection to the plane of a germ of reduced space curve is a biLipschitz homeomorphism for the outer metric. This is related to the fact that all topologically equivalent germs of plane curves are exactly the generic projections of a single space curve. The analytic algebra of this space curve is the algebra of Lipschitz meromorphic functions on any of its generic projections. An application to the geometry of polar curves is given.

Introduction

These are the lecture notes of the course given by Bernard Teissier during the second week of the " International School on Singularities and Lipschitz Geometry" which took place in Cuernavaca, Mexico from june 11 to june 22, 2018. The aim of the course was to explore the concept of "generic plane linear projection" of a complex analytic germ of curve in C N . The objects of our study will therefore be germs of curves pX, 0q Ă pC N , 0q, linear map germs π : pC N , 0q Ñ pC 2 , 0q, and the images pπpXq, 0q Ă pC 2 , 0q.

Intuitively, a projection π is generic for pX, 0q if a small variation of π does not change the "equisingularity type" (or embedded topological type) of the image pπpXq, 0q in pC 2 , 0q. The main objective was to provide algebraic criteria for a projection to be generic and to use them to prove two results related to Lipschitz geometry:

(1) That all equisingular (topologically equivalent) germs of reduced plane curves are, up to analytic isomorphism, images of a single space curve pX, 0q Ă pC N , 0q by generic linear projections π : C N Ñ C 2 , and that the restriction π|pX, 0q : pX, 0q Ñ pπpXq, 0q to pX, 0q of such a generic projection is a biLipschitz map for the metrics induced by the hermitian metrics of their 1 respective ambient spaces. In particular, all topologically equivalent germs of plane curves are biLipschitz equivalent.

(2) Given a reduced equidimensional germ of a complex space pX, 0q ãÑ pC N , 0q, with dimension d, we consider a "general" projection π : C N Ñ C 2 and the polar curve on X associated to the projection π. It is the closure in X of the critical locus of the restriction of π to the smooth part of X. If it is not empty, it is a curve usually denoted by P d´1 pX, πq which plays an important role in the study of the Lipschitz geometry of X. We can consider π as defining a plane projection of the space curve pP d´1 pX, πq, 0q which varies with π. The result is that if the projection π is sufficiently general, then it is a generic plane projection for the curve pP d´1 pX, πq, 0q Ă pC N , 0q.

The course assumed a certain familiarity with algebraic or complex analytic geometry, such as the definition of a complex analytic space X, the fact that its local algebras of functions are analytic algebras, that is, quotients of rings of convergent power series with complex coefficients, that the singular locus SingX consisting of points where the local algebra is not isomorphic to a ring of convergent power series, is a closed analytic subspace, etc. The reader is also encouraged to consult the article [Sn] of Jawad Snoussi in this volume.

What is a germ of complex analytic curve?

A complex analytic curve1 X may be locally regarded as a family of points in an open subset U of the complex affine space C N which is the union of finitely many sets of points depending analytically on one complex parameter. It can also be defined as the zero set of a finite number of holomorphic functions f 1 , . . . , f s on U satisfying certain algebraic conditions:

X " tz P U | f 1 pzq " ¨¨¨" f s pzq " 0u.

A germ of curve pX, 0q Ă pC N , 0q at a point which we take to be the origin is an equivalence class of curves in open neighborhoods of the origin. Two such objects defined respectively in U and U 1 are equivalent if their restrictions to a third neighborhood of the origin U 2 Ă U X U 1 coincide. Of course, when we speak of germs, we think of representatives in some "sufficiently small" neighborhood of the origin. Because of analyticity, to give a germ is equivalent to giving the convergent power series of f 1 , . . . , f s around the origin with respect to some coordinate system. This allows us to associate to the germ pX, 0q Ă pC N , 0q the analytic algebra of germs of holomorphic functions on pX, 0q: O X,0 :" Ctz 1 , . . . , z N u{x f 1 , . . . , f s y, where Ctz 1 , . . . , z N u denotes the ring of convergent power series. In these notes we will only be interested in reduced germs, meaning that the ideal J :" x f 1 , . . . , f s y is radical and O X,0 is a reduced analytic algebra of pure dimension 1.

In the case of plane curves pN " 2q the ideal I " x f yCtx, yu is principal and f is square free, which means that f has a factorization of the form f " f 1 ¨¨¨f r , where each f i is irreducible in Ctx, yu and they are all different. The point is that the f i 's correspond to germs pX i , 0q Ă pC, 0q of analytically irreducible curves called the branches of the curve. pX, 0q "

r ď i"1 pX i , 0q.
For arbitrary N, the branches pX i , 0q correspond to the prime ideals appearing in the primary decomposition of the ideal p0q in O X,0 p0q " P 1 X . . . X P r , where each P i is a minimal prime in O X,0

A germ of curve pX, 0q Ă pC N , 0q may also be described parametrically by r sets of power series ϕ i 1 pt i q, . . . , ϕ i N pt i q P Ctt i u, 1 ď i ď r, where again r is the number of branches. For each i, z k " ϕ i k pt i q, 1 ď k ď N defines a germ of map pD i , 0q ÝÑ pC N , 0q where D i is a disk in C. Together these r n-uples of series correspond to a multigerm of map ϕ : r ğ i"1 pD i , 0q ÝÑ pC N , 0q; z k " ϕ i k pt i q.

(1.1)

The connexion between these two definitions goes back to Newton, who showed that an equation f px, yq " 0, with f p0, 0q " 0 has solutions ypxq which are power series in x with rational exponents with bounded denominators and coefficients in the algebraic closure of the smallest field containing the coefficients of f px, yq. For Newton f px, yq is a polynomial with real coefficients, but the method works for series over any field k of characteristic zero.

Note that if

B f px, yq By

does not vanish at p0, 0q the implicit function Theorem gives us a power series ypxq with integers as exponents. In the general case, such a series ypxq " ř iPN a i x i n gives rise to a parametrization

x " t n , y "

ÿ iPN a i t i .
of one of the branches of the curve over an algebraic extension of k.

Structuring a parametrization

Suppose that we have an irreducible and reduced germ of curve in pC N , 0q, given by z k " ϕ k ptq P Cttu, k " 1, . . . , N. For simplicity we shall write z k " ϕ k ptq " ř i a piq k t i . We assume that the group generated by the exponents is Z, which means that they are coprime. Let n be the smallest exponent appearing in all the series ϕ k ptq; up to reindexing the variables z i we may assume that it is the order of ϕ 1 ptq, so that we may write ϕ 1 ptq " a pnq 1 t n p1 `ψptqq with ψp0q " 0. By making a homothetic change on the variable z 1 we may assume that a pnq 1 " 1. Since we are in characteristic zero, we may extract an nth root of the unit 1 `ψptq so that 1 `ψptq " uptq n where uptq is again invertible in Cttu. Now we make the change of parameter t 1 " tuptq so that ϕ 1 pt 1 q " t 1n . Now by making a linear change of the form z i ´ai z 1 on the coordinates z 2 , . . . , z N we may assume that z 1 is the only variable where the lowest exponent n appears. Geometrically this means that our curve is tangent to the z 1 -axis at the origin: its set-theoretic tangent cone is the z 1axis. Similarly, by making now a non linear change of coordinates of the form z i ´ř a piq k z k 1 we may assume that the first exponent appearing in each ϕ k pt 1 q is not divisible by n. This is geometrically more subtle and corresponds to Hironaka's maximal contact. Since t 1 is now our uniformizing parameter, we call it t henceforth. Let us now compare z 1 " t n with one of the other coordinates, which we may write (up to a homothetic change of variables) z i " ϕ i ptq " t b i `¨¨¨. It may be that the exponents appearing in ϕ i ptq and n are not coprime. As we shall see below it means that the projection of our curve to the pz 1 , z i q-plane is not reduced. If that is the case, we may begin by dividing all the exponents by their greatest common divisor. The interesting case is therefore that of two series expansions t n , ϕptq with coprime exponents: we are in the case N " 2 of a plane branch to which we now turn.

The case of a plane branch. As we saw, after a change of coordinates and of uniformizing parameter, we can describe our plane branch by: z 1 " t n , z 2 " ϕptq P Cttu where the smallest exponent of t in ϕptq is not divisible by n. This smallest exponent is traditionally denoted by β 1 . We take the g.c.d. of n and β 1 ; set e 1 " pn, β 1 q ă n. If e 1 " 1, the series ϕptq is of the form t β 1 `řkě1 a k t β 1 `k . If e 1 ą 1, since the exponents are coprime, there has to be a smallest exponent β 2 in the series ϕptq which is not divisible by e 1 . Then we set e 2 " pe 1 , β 2 q ă e 1 , and we continue in this manner. Since n ą e 1 ą e 2 ą ¨¨¨there exists an integer g such that e g " pe g´1 , β g q " 1. Finally we have the following structure for ϕptq: its expansion is decomposed into segments corresponding to the divisibility properties of the exponents.

z 2 " t β1 `s1 ÿ k "1 a β1`k e1 t β1`k e1 `aβ2 t β2 `s2 ÿ k "1 a β2`k e2 t β2`k e2 `¨¨¨`a β j t β j `s j ÿ k "1 a β j `k e j t β j `k e j àβg t βg `8 ÿ k "1 a βg `k t βg `k , where all a β i are ‰ 0 and each sum has to stop before the g.c.d. of the exponents drops and only the last segment is possibly infinite. The set of integers n, β 1 , β 2 , . . . , β g , which is often also denoted by β 0 , β 1 , β 2 , . . . , β g , is called the Puiseux characteristic of the branch and the β i , or sometimes the β i n , are called the characteristic exponents. It determines and is determined by the embedded topological type of the branch (see [Za1,§7], [Za3,Theorem 2.1,pg. 983],[Lej73]). This means that if two germs of plane branches pX 1 , 0q and pX 2 , 0q have the same Puiseux characteristic there exists a homeomorphism pU 1 , 0q Ñ pU 2 , 0q of neighborhoods of the origin mapping the representative X 1 Ă U 1 to X 2 Ă U 2 , and conversely. The two germs are also said to be equisingular. We shall meet this Puiseux characteristic again after Example 1.4.25 below, where we shall see that it determines not only the topology but also the biLipschitz geometry of the branch.

After what we have seen, the expansion above can be reinterpreted as a Newton expansion in terms of t " z 1 n

1 , but here we have to choose a n-th root of z 1 . The algebraic interpretation is that ϕpz

1 n 1 q P Cttz 1 uutz 1 n
1 u determines a cyclic extension of the field Cttz 1 uu of meromorphic functions in z 1 with Galois group equal to the group µ n of n-th roots of 1. The n series ϕpωz 1 n 1 q, ω P µ n , are the roots of a unitary polynomial ś ωPµ n `z2 ´ϕpωz

1 n 1 q ˘P Ctz 1 urz 2 s whose vanishing is an equation for our germ of curve in the sense we shall see in the next section. The structure of the series gives rise to a filtration of the Galois group:

µ n Ą µ e 1 Ą µ e 2 Ą ¨¨¨Ą µ e g " t1u,
with the characteristic property that if we set n " e 0 and denote by ν t the t-adic order of a series, then for 1 ď k ď g, we have that ω P µ e k ´1 zµ e k ðñ ν t pϕpωtq ´ϕptqq " β k .

Let us now refine the structure according to [Za2,Chapters III,IV,V]. The parametrization of a branch by t n , yptq as above presents its analytic algebra O X,0 as a subalgebra of Cttu. The t-adic orders of the series in t which are in O X,0 form a numerical semigroup Γ Ă N since one can multiply them and stay in O X,0 . Since the exponents are coprime the complement of Γ in N is finite (Dickson's Lemma) and the semigroup Γ is finitely generated. The smallest element c of N such that all integers ě c are in Γ is called the conductor of the semigroup. It is not difficult to verify (see [Za2,Chapter III,Lemma 1.1]) that if the order of a series ξptq P O X,0 is ą β 1 , then ξptq P xx, yy 2 , and therefore if the order s of ξptq is in Γ we can make a change of coordinates x 1 " x, y 1 " y ´ξptq to eliminate a term in t s from the expansion of yptq. Using this, and the fact that by definition any element of Γ is the order of a series in O X,0 , Zariski proved in [Za2, Chapter III, Proposition 1.2]: Proposition 1.0.1 (Zariski) 1) Assume that n ą 2. Then one has c ě β 1 `1. Let s 1 , . . . , s q be the integers of the set t β 1 `1, . . . , cu which do not belong to Γ. The branch pX, 0q is analytically isomorphic to a branch given parametrically by:

x 1 ptq " t n , y 1 ptq " t β 1 `q ÿ i"1 a 1 s i t s i .
2) If n " 2 then β 1 is odd since our germ is irreducible and the conductor is β 1 ; our curve is analytically isomorphic to xptq " t 2 , yptq " t β 1 .

Zariski calls this a short representation. There are more simplifications of the expansion of yptq one can make without changing the analytic type. See [Za2,Chapters III,IV,V].

The next thing we need to know is that the semigroup Γ determines and is determined by the Puiseux characteristic of the branch: it is a complete invariant of the equisingularity class. See [Za2,Chap. II,§3]. In particular, in the short expansion, the coefficients of the t β i are ‰ 0.

With this description of branches, we are able to describe the contact of two branches, which plays a key role in the characterization of the topological (and biLipschitz) type of a reduced germ of plane curve.

We shall see below how, conversely, the image of a parametrization can be defined by equations. The modern presentation of the parametrization of a curve goes through the normalization, which is the topic of the next section.

Normalization

The property of being normal has an algebraic aspect which has to do with integral extension of rings.

Definition 1.1.1 Let R Ă S be rings.

• The inclusion R Ă S is called a finite extension if S is a finitely generated R´module. Suppose that R is a reduced ring. Recall that the set of non-zero divisors of a ring R is a multiplicatively closed set and the corresponding ring of fractions QpRq is called the total ring of fractions. It has the property that the canonical morphism R Ñ QpRq is injective.

The normalization of R is defined as the set R of all elements of QpRq which are integral over R. It is a reduced ring, integrally closed in QpRq and whose total ring of fractions coincides with QpRq. In particular, the normalization R is a normal ring. Moreover, for the rings appearing in analytic or algebraic geometry, the extension R Ă R is finite in the sense that R is a finitely generated R-module. 2 So what about if we start with the analytic algebra O X,0 of a germ of analytic space pX, 0q Ă pC N , 0q? We will say that the germ pX, 0q is normal if O X,0 is a normal ring.

2 It is interesting to note that the term "integral" comes from algebraic number theory in the tradition of Dedekind and the definition of the ring of integers of an algebraic number field, while the term "normal" was used by Zariski (see [Za]) in the course of his studies in birational geometry and resolution of singularities to designate an algebraic variety which could not be presented as the image of a different one by a finite birational map. This is why the terms "integral closure in the total ring of fractions" and "normalization" are used in algebraic or analytic geometry as names for the algebraic and geometric aspects of the same operation.

• Unique factorization domains are normal ([De-P00, Thm 1.5.5]) so the ring of power series Ctz 1 , . . . , z n u and the corresponding smooth germ pC N , 0q are normal. • Noetherian normal local rings are integral domains ([De-P00, Thm 1.5.7]), so a normal germ pX, 0q is irreducible. • Suppose pX, 0q is irreducible. Since O X,0 and its normalization have the same total ring of fractions, which in this case is a field, it follows from what we have just seen that O X,0 is a local noetherian domain. Moreover, by Cor. 3.325] it is an analytic algebra and so we can associate to it a normal germ pX, 0q. In particular we have:

O X,0 " O X,0 .
• Splitting of normalization ( Thm. 1.5.20]) tells us that that if we have the irreducible decomposition

pX, 0q " pX 1 , 0q Y . . . Y pX s , 0q,
then the normalization O X,0 is equal to a direct sum of analytic algebras which are the normalizations of the analytic algebras O X i ,0 corresponding to the irreducible components pX i , 0q:

O X,0 " s à i"1 O X i ,0 .
Note that this implies that pX, 0q and pX, 0q have the same dimension.

A multi-germ of analytic spaces pX, xq is a finite disjoint union:

pX, xq :" pX 1 , x 1 q \ pX 2 , x 2 q \ . . . \ pX r , x r q of germs of analytic spaces. The ring O X, x by definition is equal to

À r i"1 O X i , x i . The multigerm pX, xq is called normal if O X, x is a normal ring.
Let pY, yq " pY 1 , y 1 q \ . . . \ pY s , y s q be another multi-germ. A map ϕ : pX, xq Ñ pY, yq of multi-germs is given by a system of maps ϕ i : pX i , x i q Ñ pY αpiq , y αpiq q, i P t1, . . . , ru, αpiq P t1, . . . , su. With this definition at hand, for any germ of analytic space pX, 0q with irreducible decomposition pX, 0q " pX 1 , 0q Y . . . Y pX s , 0q, we can now obtain a normal multigerm pX, xq " pX 1 , x 1 q \ . . . \ pX s , x s q with associated normal ring

O X,0 " s à i"1 O X i ,0 " s à i"1 O X i , x i ,
and it is not hard to prove that the inclusion map O X,0 ãÑ O X,0 induces a finite and generically 1-1 map, proving thus the existence of normalization ( Thm 4.4.8]). Note that, geometrically, the normalization of a germ separates the irreducible components and normalizes each of them separately.

Example 1.1.3 Let pX, 0q Ă pC 2 , 0q be the germ of plane curve defined by f px, yq " x 2 ´y2 . It has two irreducible components pX 1 , 0q and pX 2 , 0q with associated analytic algebras

O X 1 ,0 " Ctx, yu{xx ´yy O X 2 ,0 " Ctx, yu{xx `yy.
These two germs are smooth, in particular they are normal and we have:

O X,0 " Ctx, yu xx 2 ´y2 y ÝÑ Ctx, yu xx ´yy à Ctx, yu xx `yy " O X,0 f Þ ÝÑ p f `xx ´yy, f `xx `yyq
Since the germs are smooth and of dimension 1, their analytic algebras are isomorphic to the ring of convergent power series Cttu:

Ctx, yu{xx ´yy Ñ Cttu x Þ Ñ t, y Þ Ñ t Ctx, yu{xx `yy Ñ Ctuu x Þ Ñ u, y Þ Ñ ´u
This means that the resulting normalization map

n : pC, 0q \ pC, 0q Ñ pX, 0q
is the parametrization of each of the branches t 1 Þ Ñ pt, tq and t 2 Þ Ñ pu, ´uq.

It is useful to consider a function-theoretic interpretation of normal spaces. A general result tells us that in a smooth germ pC d , 0q if you have a meromorphic function which is (locally) bounded then it is actually holomorphic (See for example IV.4]). The algebraic version is that a locally bounded meromorphic function h satisfies an integral dependence relation of the form:

h m `c1 h m´1 `¨¨¨`c m " 0; c j P O n :" Ctz 1 , . . . , z n u,
and since O n is normal then h P O n . Now there are many more analytic spaces for which O X, x is normal than just the non singular ones.

Definition 1.1.4 Given a reduced germ of analytic space pX, xq, we call a function f : XzSing X Ñ C weakly holomorphic at x P X if :

• f is holomorphic on XzSing X in a neighborhood of x.

• f is (locally) bounded near x.

A function is weakly holomorphic on X if it is so at every point.

The key point is proving that the germs at x P X of weakly holomorphic functions on X form a ring which is canonically isomorphic to the normalization of O X, x . That is, f is weakly holomorphic on X if and only if it is meromorphic and satisfies an integral dependence relation. This gives us the following characterization:

Theorem 1.1.5 Thm 4.4.15] 1) Let pX, xq be a germ of reduced analytic space. Then a function f is weakly holomorphic on X if and only if f is in the integral closure of O X, x in its total ring of quotients.

2) The integral closure of O X, x in its total ring of quotients is a direct sum of analytic algebras.

3) The reduced germ pX, xq is normal if and only if every weakly holomorphic function germ can be extended to a holomorphic function.

Remark 1.1.6 Since this fact is fundamental for what follows, here is an idea of why boundedness and polynomial equation are related: The roots of a polynomial are bounded in terms of its coefficients, so a solution of a polynomial equation with holomorphic coefficients is bounded because holomorphic functions are. In the other direction, let h " f g , with f , g P m pX,0q be our meromorphic function, let pY, 0q Ă pX, 0q be the subset defined by the ideal x f , gyO pX,0q , and consider the analytic subspace X 1 of X ˆP1 pCq which is the closure of the graph of the map XzY Ñ P 1 pCq defined by x Þ Ñ p f pxq : gpxqq P P 1 pCq. It is contained in the hypersurface of X ˆP1 pCq defined by T 2 f pxq ´T1 gpxq " 0 where pT 1 : T 2 q are projective coordinates on P 1 . The first projection induces a holomorphic map e : X 1 Ñ X (we are blowing-up the ideal x f , gy). The fiber over 0 is a complex analytic subspace of P 1 pCq and therefore is either P 1 pCq or a finite subset of it. If our meromorphic function is bounded, the point p1 : 0q P P 1 pCq is not in the fiber, so that by the Weierstrass preparation Theorem (see Theorem 1.1.8 below), for a small enough representative X of the germ pX, 0q the map X 1 Ñ X is finite and X 1 has to be a hypersurface in X ˆC: its equation is our integral dependence relation.

Example 1.1.7 For the germ pX, 0q Ă pC 2 , 0q defined by xy " 0 we have O X,0 " Ctxu ' Ctyu.

The function f " p1, 0q, meaning it is the constant function 1 on the x axis and the constant function 0 on the y axis, is holomorphic on XzSing X " Xzt0u and is certainly bounded so it is weakly holomorphic. Note that it can not be continuously extended to pX, 0q. As a meromorphic function it can be written as f px, yq " x x `y .

Let us wrap up this discussion on normal spaces and normalization by stating several important properties of which you can find detailed expositions in

[Loj91], [G-L-S07] and [Kau83].
1. If X is reduced, the non normal locus is the set of points x P X where the local algebra O X, x is not normal; it is the complement of the normal locus and is a closed analytic subspace contained in the singular locus SingX of X. It is defined by the conductor sheaf which is the annihilator of the coherent O X -module O X {O X and thus a coherent sheaf of ideals. 2. If T is a normal space and X is reduced then any map T Ñ X which does not map any irreducible component of T to the non-normal locus of X factors uniquely through the normalization n : X Ñ X. 3. If X is normal then dim SingpXq ď dim X ´2 (Singular locus of codimension at least 2). 4. If X is normal, the polar locus of a meromorphic function is either of codimension 1 or empty.

Going back to the curve case, a classical result of commutative algebra ([De-P00, Thm 4.4.9]) states that a Noetherian local ring of dimension one is normal if and only if it is regular. This implies that if pX, 0q " Ť r i"1 pX i , 0q Ă pC N , 0q is a germ of analytic curve with r branches then the normal ring O X,0 is isomorphic to a direct sum of r copies of Cttu and the corresponding normalization map is equal to the parametrization of each branch, thus recovering the description in (1.1). For plane curves, this result can also be seen using algebraic field extensions, but first we need a couple of definitions and the Weierstrass preparation Theorem. A convergent power series f P Ctz 1 , . . . , z N u is called regular of order b in z N if the power series f p0, . . . , 0, z N q in the variable z N has a zero of order b. A simple calculation shows that if f is of order b in the sense that f P xz 1 , . . . , z N y b zxz 1 , . . . , z N y b`1 , then after a general linear change of coordinates, f is regular of order b in z N (see Lemma 3.2.2]). Geometrically this means that if we consider the germ of hypersurface pX, 0q Ă pC N ´1 ˆC, 0q defined by f and the first projection p : X Ñ C N ´1, then for a small enough representative the fiber p ´1p0q is the single point 0.

Theorem 1.1.8 (Weierstrass Preparation Theorem) (see Thm 3.2.4]) Let f P Ctz 1 , . . . , z N u be regular of order b in z N . Then there exists a unique monic polynomial

P P Ctz 1 , . . . , z N ´1urz N s Ppz 1 , . . . , z N q " z b N `a1 pz 1 , . . . , z N ´1qz b´1 N `¨¨¨`a N pz 1 , . . . , z N ´1q
with a i p0q " 0, and a unit u P Ctz 1 , . . . , z N u such that we have the equality of convergent power series f " uP.

As a consequence of this result we deduce two important facts: if we choose adequate coordinates such that f " uP then it is equivalent to seek solutions of f pz 1 , . . . , z N q " 0 and of Ppz 1 , . . . , z N q " 0. As a geometric consequence of this we get that if we consider the first projection as before and p ´1p0q " t0u, then for any point q " pq 1 , . . . , q N ´1q P C N ´1 sufficiently close to the origin the points of the fiber p ´1pqq correspond to the roots of the polynomial of degree b Ppq 1 , . . . , q N ´1, z N q " z b N `a1 pq 1 , . . . , q N ´1qz b´1 N `¨¨¨`a N pq 1 , . . . , q N ´1q, and so all nearby fibers are also finite. More generally one uses this result to prove that if a complex analytic map p : X 1 Ñ X is such that for some point 0 P X we have that p ´1p0q is a finite set, then there exists a neighborhood U of 0 in X such that the restricted map p ´1pU q Ñ U is finite. See Thm 3.4.24].

For curve singularities, there is a classical invariant which measures how far the singularity is from being normal, or non singular. It has several geometric interpretations, the classical one being "diminution of genus", and we shall see more about it below.

Definition 1.1.9 Let pX, 0q be a reduced curve singularity. Its δ invariant is

δ " dim C O X,0 O X,0 .
This quotient is a finite dimensional vector space because it is the stalk of a coherent sheaf supported at the origin. For plane, and more generally Gorenstein, branches we have the equality c " 2δ, where c is the conductor defined before Proposition 1.0.1. See [Za2,Chap. II,§1].

Going back to the plane curve case, that is curves pX, 0q Ă pC 2 , 0q defined by a convergent power series f P Ctx, yu, or according to the Weierstrass preparation Theorem and possibly after a linear change of coordinates, by a polynomial P P Ctxurys. Now from an algebraic point of view, consider the field of fractions Cttxuu of the integral domain Ctxu; the irreducible polynomial y n ´x P Cttxuurys defines an algebraic extension of degree n of Cttxuu, denoted by Cttx 1 n uu, which is a Galois extension with Galois group equal to the group µ n of n-th roots of unity in C. The action of µ n is exactly the change in determination of x 1 n determined by x 1 n Þ Ñ ωx 1 n for ω P µ n . A series of the form y " ř a i x i n such that the greatest common divisor of n and all the exponents i which effectively appear is 1 gives n different series as ω runs through µ n .

Suppose now that our polynomial P is an irreducible element of Ctxurys of degree n. Then the Newton polygon method (see for example [Tei07], [Che78], or [Br-K86, Section 8.3]) provides a series ypx 1{m q P Ctx 1 n u such that Ppx, ypx 1 n qq " 0 and we have the equality:

Ppx, yq " ź ωPµ n ´y ´ypωx 1 n q ¯.
In particular we have that

Cttxuu ˚:" ď nPN Cttx 1 n uu
is an algebraically closed field (See [Wal78, IV.3] or [Che78,Thm 8.2.1]), and so every polynomial P P Ctxurys has all its roots in Cttxuu ˚. Finally, the relation with the parametrizations given by the normalization is the following, if ypx

1 n q P Ctx 1 n u Ă Cttxuu is a root of Ppx, yq, then by taking x " t n we get the parametrization t Þ Ñ pt n , yptqq.
Let us finish this section by looking at plane projections from an algebraic perspective. For simplicity suppose pX, 0q Ă pC N , 0q is a reduced and irreducible germ of complex analytic curve with N ě 3. Let us write the associated analytic algebra

O X,0 " Ctz 1 , . . . , z N u I ,
where I is a prime ideal, and so O X,0 is an integral domain. If we choose a sufficiently general coordinate system (or if you prefer after a general linear coordinate change) the Noether normalization Theorem ([De-P00, Corollary 3.3.19]) tells us that we have a finite ring extension Ctz 1 u ãÑ O X,0 . This implies that the we have an algebraic field extension

Cttz 1 uu Ă Quot pO X,0 q ,
and by the primitive element Theorem there exists an element f P O X,0 such that Quot pO X,0 q " Cttz 1 uur f s.

So if we denote by Ctz 1 , f u the analytic algebra obtained as the quotient of Ctx, yu by the kernel of the map Ctx, yu Ñ O X,0 defined by x Þ Ñ z 1 `J, y Þ Ñ f then we have finite ring extensions with the same field of fractions

Ctz 1 , f u ãÑ O X,0 ãÑ Cttu.
Now Ctz 1 , f u is the analytic algebra of a plane curve pX 1 , 0q Ă pC 2 , 0q and it has the same normalization as O X,0 . We have used the primitive element Theorem as a substitute for the proof of the existence of a projection C N Ñ C 2 sufficiently general for it to induce a " bimeromorphic" map pX, 0q Ñ pX 1 , 0q. However the primitive element Theorem does not tell us much about the geometric nature of the projection. That is the object of the following sections.

Fitting Ideals -A good structure for the image of a finite map

In this section, following [Tei73, §3] and [Rim72, Definition 5.6], [Tei77, §5], we will give the definitions of Fitting ideals, which we will use later to give a definition of the image, as a complex analytic space, of a finite map between complex analytic spaces.

Let A be a ring, and let M be an A-module of finite presentation, that is, there is an exact sequence, called a presentation of M:

A q Ψ ÝÑ A p ÝÑ M ÝÑ 0,
where p, q P N. For each integer j we associate to M the ideal F j pMq of A generated by the pp ´jq ˆpp ´jq minors of the matrix (with entries in A) representing Ψ. Here we need the convention that if there are no pp jq ˆpp ´jq minors because j is too large, i.e., j ě p, then F j pMq " A (the empty determinant is equal to 1) and if, at the other extreme, p ´j ą q, set F j pMq " 0 (the ideal generated by the empty set is 0).

A Theorem of Fitting (see [To72, Chap. I, §2], [Eis95, Chap. 20, §2]) asserts that the ideals F j pMq depend only on the A-module M and not on the choice of a presentation. We call F j pMq the j-th Fitting ideal of M.

More generally, if pX, O X q is a ringed space, and M a coherent sheaf of O X -modules, we can define a sheaf of ideals F i pMq of O X , by defining F i pMq locally as above, and then by uniqueness the ideals found locally patch up into a sheaf of ideals. Remark also that since F i pMq is locally finitely generated, F i pMq will be a coherent sheaf of ideals as soon as O X is coherent, which is the case for a complex analytic space by Oka's Theorem (see [Loj91, Chap.

VI, 1.3]).

Let now f : pX, O X q Ñ pY, O Y q be a map of complex analytic spaces. We would like to define the image of f as a complex analytic subspace of pY, O Y q. This is not always possible, and in particular if one hopes to get a closed complex subspace of Y it is better to assume f is proper, and here we will consider only the case where f is finite (that is, proper with finite fibres).

The first sheaf of ideals that comes to mind as a candidate to define f pXq is the sheaf of functions g on Y such that g ˝f " 0 on X, i.e., the annihilator sheaf of the sheaf of O Y -modules f ˚OX , which is coherent by a theorem of Grauert-Remmert:

Ann O Y p f ˚pO X qq " sheaftfunctions g on Y such that g ¨f˚OX " 0 u.
This is not a good choice because its formation does not commute with base extension, as we will show by an example below (Example 1.2.3).

The second option is the 0th Fitting ideal of f ˚OX , which set theoretically also defines the image of f , since as a set the subspace of Y defined by it is ty

P Y | dim C p f ˚OX q ą 0u " ty P Y | p f ˚OX q y ‰ 0u. Indeed, since the O Y -module f ˚OX is coherent, it has locally on suitable open sets U of Y a presentation by an exact equence of O Y pUq-modules: O Y pUq q Ψ ÝÑ O Y pUq p ÝÑ f ˚OX pUq ÝÑ p0q.
The sheaf of ideals F 0 p f ˚OX q is then generated on U by the p ˆp minors of a matrix representing Ψ.

Since both the formation of direct images and the formation of Fitting ideals commute with base change (see proposition 1.2.2 below), this definition of the image will also have this property. So we set: Definition 1.2.1 Let f : X Ñ Y be a finite morphism of complex analytic spaces. The image imp f q of f is the subspace of Y defined by the coherent sheaf of ideals F 0 p f ˚OX q. It is sometimes called the Fitting image of f to distinguish it from the one defined by the annihilator.

Proposition 1.2.2 1. The formation of imp f q commutes with base change: Given a complex analytic map φ : T Ñ Y , consider the map f T : X ˆY T Ñ T obtained by base extension, where X ˆY T is the fiber product. Then imp f T q " φ ´1pimp f qq as analytic spaces. 2. We have the inclusion F 0 p f ˚OX q Ă Annp f ˚OX q and the equality a

F 0 p f ˚OX q " a Annp f ˚OX q. Proof 1) Since O X is a finitely generated O Y -module the O T -module O X ˆY T is equal to O X b O Y O
T and if M is a finitely presented A-module as above and A Ñ B is a map of algebras, then

B q Ψb A 1 ÝÑ B p ÝÑ M b A B ÝÑ 0 is a presentation of M b A B as a B-module and the matrix of Ψ b A 1 is the matrix of Ψ so that F j pM b A Bq " F j pMq.B.
2) The inclusion follows directly from Cramer's rule and the equality from the definition of the Fitting ideal as defining the set of points where the cokernel of the second arrow is not zero. Example 1.2.3 Let f : pC, 0q Ñ pC 2 , 0q be given by x " t 2k , y " t 3k for some integer k. The set-theoretic image of f is the curve y 2 ´x3 " 0. However, we wish to obtain an ideal defining a space supported on that curve, but possibly with nilpotent functions. Let us compute F 0 p f ˚pO C q 0 as the 0-th Fitting ideal of Cttu considered as Ctx, yu-module via the map of rings Ctx, yu Ñ Cttu sending x to t 2k and y to t 3k . We must write a presentation of Cttu as Ctx, yumodule. Let e 0 " 1, e 1 " t, . . . , e 2k´1 " t 2k´1 . It is easily seen that they form a system of generators of Cttu as Ctx, yu-module, and that between them we have the following 2k relations:

xe k ´ye 0 " 0, x 2 e 0 ´ye k " 0 xe k`1 ´ye 1 " 0, x 2 e 1 ´ye k`1 " 0 . . . . . . xe 2k´1 ´ye k´1 " 0,
x 2 e k´1 ´ye 2k´1 " 0 which are independent. Hence we have a sequence of Ctx, yu-modules:

0 ÝÑ 2k´1 à i"0 Ctx, yue i ψ Ý Ñ 2k´1 à i"0 Ctx, yue i ϕ Ý Ñ Cttu ÝÑ 0
with ϕpe i q " t i , and ψ is given by the 2k ˆ2k matrix

ψ " » - - - - - - - - - - - - - ´y 0 ¨¨¨0 x 0 ¨¨¨0 0 ´y ¨¨¨0 0 x ¨¨¨0 . . . . . . . . . . . . 0 0 0 ¨¨¨´y 0 0 ¨¨¨x x 2 0 ¨¨¨0 ´y 0 ¨¨¨0 0 x 2 ¨¨¨0 0 ´y ¨¨¨0 . . . . . . . . . . . . 0 0 0 ¨¨¨x 2 0 0 ¨¨¨´y fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl
It is not hard to see that the sequence is exact, which means that the independent relations we have found must generate all relations between the e i . Indeed, there is a general reason why Cttu must have a resolution of length 1 as Ctx, yu-module: the Ctx, yu-module Cttu is of homological dimension one (see ) and therefore the module of relations between the e i is a free submodule of À 2k´1 i"0 Ctx, yu and thus of rank ď 2k ´1. By permuting rows and columns of ψ one checks that detpψq " py 2 ´x3 q k i.e., we have shown that

F 0 p f ˚OC q 0 " py 2 ´x3 q k Ctx, yu
Let us now calculate Ann Ctx, yu Cttu; the annihilator is just the kernel of the map Ctx, yu Ñ Cttu, which is the ideal generated by py 2 ´x3 q, certainly different from our Fitting ideal if k ą 1.

Let us now make a base change by restricting our map over the x-axis, i.e., by the inclusion ty " 0u Ă pC 2 , 0q or algebraically by Ctx, yu Ñ Ctxu sending y to 0. Then the annihilator of Cttu  Ctx, yu Ctxu " Cttu{pt 3k q viewed as Ctxu-module is px 2 qCtxu while the image in Ctxu of py 2 ´x3 qCtx, yu is px 3 qCtxu. This shows that the formation of the annihilator does not commute with base change.

Equations versus Parametrizations

As we said in subsection 1.0.1, a germ of curve pX 0 , 0q, abstractly, is a germ of a purely 1-dimensional analytic space, hence it is described by an analytic algebra O X 0 ,0 of pure dimension 1. Geometrically, pX 0 , 0q can be effectively given in two ways:

By equations: By giving an ideal I " x f 1 , . . . , f m y in Ctx 1 , . . . , x N u such that O X 0 ,0 » Ctx 1 , . . . , x N u{I. Saying that O X 0 ,0 is purely one-dimensional means that the ideal x0y has a primary decomposition x0y " Q 1 X . . . X Q r where ? Q i " P i is a minimal prime ideal in O X 0 ,0 , and dimpO X 0 ,0 {P i q " 1 for i " 1, . . . , r.

By a parametrization: By giving ourselves a germ of finite map p :

Ů r i"1 pC, 0q Ñ pC N , 0q.
Here one has to be very careful: except when n " 2, it is not true, even if r " 1 and p is generically 1-to-1 so that the image (given by the Fitting structure) of this mapping is a reduced curve: it will have "embedded components" concentrated at the singular points, as will be shown in Example 1.2.4. The analysis of this phenomenon is beyond the scope of these notes. The case where n " 2 is explained in Proposition 1.2.6 in the next section.

Example 1.2.4 Consider the curve pX 0 , 0q parametrized by nptq " pt 4 , t 6 , t 7 q which is a complete intersection (with the reduced structure) with ideal xy 2 ´x3 , z 2 ´x2 yyCtx, y, zu.

We have that Cttu is generated as a Ctx, y, zu-module by e 0 " 1, e 1 " t, e 2 " t 2 and e 3 " t 3 and it is not difficult to see that the relations are described by the following matrix

Ψ " » - - - - - - - - - - - y 0 ´x 0 0 y 0 ´x ´x2 0 y 0 0 ´x2 0 y z 0 0 ´x ´x2 z 0 0 0 ´x2 z 0 0 0 ´x2 z fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl
that is, Ψ is the matrix of a presentation Ctx, y, zu 8 Ψ ÝÑ Ctx, y, zu 4 ÝÑ Cttu ÝÑ 0.

of the Ctx, y, zu-module Cttu. Computing the 4 ˆ4 minors of Ψ we find that:

F 0 pCttuq " xy 2 ´x3 , z 2 ´x2 yy X xz 2 , x y 3 , y 4 , x y 2 z x4 z, y 3 z, x 4 y, x 3 y 2 , x 3 yz, x 6 , x 5 zyCtx, y, zu,
where a xz 2 , x y 3 , y 4 , x y 2 z ´x4 z, y 3 z, x 4 y, x 3 y 2 , x 3 yz, x 6 , x 5 zy " xx, y, zyCtx, y, zu. The ring Ctx, y, zu{F 0 pCttuq is not purely 1-dimensional: it has an embedded component: an ideal of the primary decomposition of the ideal p0q which defines a subspace of strictly lower dimension, in this case dimension zero.

Deformations of equations vs. deformations of parametrizations

In this subsection we consider deformations of a curve. We will follow the presentation given in . The results in this subsection are due to B. Teissier (see [Tei77]). Let pX 0 , 0q Ă pC N , 0q be a germ of a reduced curve and X 0 Ă B 0 a representative, where B 0 Ă C N is a small open ball with center 0. Let

ϕ 0 : X 0 " r ğ j"1 D j Ñ X 0 Ă B 0 ,
ϕ 0 ptq " pϕ 0 pt 1 q, . . . , ϕ 0 pt r qq , t " pt 1 , . . . , t r q, be a representative of the normalization of X 0 , where r ğ j"1 D j is the disjoint union of r open discs centered at the origin in C, ϕ 0 is a r-uple of distinct maps ϕ 0 |D j : pD j , 0q Ñ B 0 , and for each j the restriction ϕ 0 |D j is a homeomorphism pD j , 0q Ñ pX j 0 , 0q, where pX j 0 , 0q is the j-th branch of pX 0 , 0q. It induces an analytic isomorphism pD j , 0qzt0u Ñ pX j 0 , 0qzt0u because an analytic map C Ñ C which is a homeomorphism is an isomorphism. Definition 1.2.5 Let D Ă C q be a small disc with center 0. A deformation of the normalization of X 0 is a holomorphic mapping ϕ : X 0 ˆD " r ğ j"1 pD j ˆDq Ñ B 0 , such that ϕpt, vq " ϕ 0 ptq`vψpt, vq, t P X 0 , v P D and ψpt, vq " pψpt 1 , vq, . . . , ψpt r , vqq with ψpt j , vq : pD j ˆD, 0q Ñ B 0 . Note that we are dealing with representatives of germs of deformations of germs.

Then for sufficiently small D j and D we have that φ " pϕ, vq : X 0 ˆD Ñ B 0 ˆD is a finite mapping and therefore

X " φpX 0 ˆDq Ă B 0 ˆD is a q `1-dimensional analytic subset.
Proposition 1.2.6 Given a germ of finite map n : Ů r i"1 pD j ˆD, 0q Ñ pC 2 , 0q ˆD with D Ă C q as above, corresponding to a map of analytic Calgebras O C q ,0 tx, yu Ñ À r i"1 O C q ,0 tt i u which is the identity on O C q ,0 and makes the second algebra a finite module over the first, the Fitting ideal Theorem B.8.11]) of dimension q `1, so by the Auslander-Buchsbaum formula its homological dimension is one (see (see [G-L-S07, Theorem B.9.3]), which implies that its minimal presentations are exact sequences of O C q ,0 tx, yu-modules

F 0 p À r i"1 O C q ,0 tt i uq is a non zero principal ideal of O C q ,0 tx, yu. Proof The argument goes back to [Tei73, Chap.III, 3.4] (see also [Tei77, §5], [G-L-S07, Exercise 1.6.4], [Mo-P89, Proposition 3.1]): the depth of the O C q ,0 tx, yu-module À r i"1 O C q ,0 tt i u is q `1 because it is Cohen-Macaulay (see [G-L
p0q Ñ O C q ,0 tx, yu p Ñ O C q ,0 tx, yu p Ñ r à i"1 O C q ,0 tt i u Ñ p0q.
Therefore the 0-th Fitting ideal is generated by the determinant of a p ˆpmatrix.

See also [Mo-P89, Lemma 2.1].

Applying Proposition 1.2.6 with q " 0, we see that none of the plane images we consider has embedded components. The Fitting image of a parametrization is reduced if and only if for i " 1, . . . , r the set of all exponents appearing in the series xpt i q or in ypt i q is coprime. For given i, the gcd of those exponents is the degree of the map pD, 0q Ñ pC 2 , 0q defined by xpt i q, ypt i q and it is also the degree at which the equation f i px, yq " 0 of the reduced image curve is raised to give a generator of the Fitting ideal of the Ctx, yu-module Ctt i uq.

Compare with Example 1.2.3 and see [Mo-P89, Proposition 3.1] for a more general result.

Going back to the case q ą 0, we see that if the parametrization for v " 0 has a reduced image X 0 , then the hypersurface X in D ˆC2 which is the image of the parametrization is reduced. Otherwise, since it is the Fitting image, applying the compatibility with base change to 0 P D, we find that the special fiber X 0 would not be reduced. But then each fiber is reduced and its parametrization is an isomorphism outside of the singularities; it is a bimeromorphic map. This implies that the map p : Ů r i"1 pD j ˆD, 0ˆ0q Ñ pX, 0q is the normalization of the hypersurface X because the source is normal, and the map p is finite and bimeromorphic. It is a non singular normalization of the hypersurface X which induces the normalization of each fiber of the projection map X Ñ D. So we have proved:

Corollary 1.2.7 The parametrization of the total space of the family of plane curves n : pX, 0q Ñ pD, 0q obtained by deforming the parametrization of the germ of reduced plane curve pX 0 , 0q is a simultaneous normalization.

Moreover, we can observe that the O C q ,0 -modules O X,0 and O X,0 " À r i"1 O C q ,0 tt i u are flat. The first one because a hypersurface is Cohen-Macaulay (see Theorem B.8.11]) and the second one because it is a sum of flat modules (see [G-L-S07, Corollary I 1.88]). We have the exact sequence of O C q ,0 -modules:

p0q Ñ O X,0 Ñ O X,0 Ñ O X,0 O X,0 Ñ p0q,
and the flatness of the first two modules implies that of the third. Now by the Weierstrass Preparation Theorem, the singular locus of X is finite over D, 0q and is the support of the O C q ,0 -module O X,0 O X,0 , which is flat and thus locally free. So the dimensions of its fibers over points v P D is constant. Provided D and B 0 are sufficiently small, this dimension is the sum of the δ invariants (see Definition 1.1.9) of the finitely many singularities of the curve X v " n ´1pvq, which all tend to ) as v Ñ 0. So we have: Corollary 1.2.8 In a small enough representative of family of reduced plane curves obtained by deformation of the parametrization of the special fiber, the sum of the δ invariants of the singularities of the fibers is constant.

We shall use this in the proof of Proposition 1.3.6 below.

We have just seen that the space X defined by deforming a parametrization of a plane curve can also be described as the space defined by deforming an equation for the reduced plane curve X 0 , since the equation for X has to reduce to the equation for X 0 when setting v " 0. The situation is more delicate for parametrized curves in C N with N ą 2, not only because of the behavior of the Fitting ideal, but also because in general deforming equations does not produce a flat family. The general definition of a deformation of a germ pX 0 , 0q Ă pC N , 0q is a germ pX, 0q Ă pC N , 0q ˆpS, 0q, flat over S and defined by equations which, when restricted over 0 P S give a set of equations of X 0 , up to isomorphism. It is therefore a reasonable question to ask whether any such deformation can be obtained by a deformation of the parametrization of X 0 in the sense that X is the reduced image of such a deformation.

The answer is a converse to Corollary 1.2.8, as follows:

Proposition 1.2.9 (See [Tei80, Theorème 1, page 80], and [G-L-S07, §2.6] for plane curves) Let p : pX, 0q Ñ pS, 0q be a flat morphism where all the fibers are reduced curves and S is non singular. Then, for suitable representatives, the following conditions are equivalent:

• The normalization X is non singular, the composed map X n Ñ X p Ñ S is flat and for each s P S the map of fibers pXq s Ñ X s is the normalization.

• The sum δ s of the δ invariants of the singular points of the fibers X s is independent of s P S.

Since the map p ˝n is flat with non singular fiber, at every point of X lying above 0 P X, the space X is locally isomorphic over S to a product of a disk by S. This shows that the map n is a deformation of the parametrization of X 0 . The assumption that the fibers are reduced is necessary, as evidenced by the following example.

Example 1.2.10 Consider the of curve X 0 in C 3 given by the equations x " 0, z 2 ´y3 " 0. The normalization of X 0 is given by ϕptq " p0, t 2 , t 3 q.

Consider the deformation Φpv, tq " pvt, t 2 , t 3 , vq. So, the reduced image X r ed of Φ is given by the following equations:

x 2 ´v2 y " 0, x y ´vz " 0, xz ´vy 2 " 0, z 2 ´y3 " 0.

Now, when we consider the projection f : X r ed Ñ D, the fiber f ´1p0q " X 0 is given by

x 2 " 0, x y " 0, xz " 0, z 2 ´y3 " 0.
Note that it is not reduced at the origin, hence there is no deformation of the equations x " z 2 ´y3 " 0 which defines X r ed . One can understand this as follows: while the special fiber of our family of curves has embedding dimension two, the general fiber has embedding dimension three. In an analytic family the embedding dimension of the fibers can only increase by specialization so that in our analytic family f : X r ed Ñ D the ideal defining the special fiber has in its primary decomposition an infinitesimal embedded component with ideal xx 2 , y, zy sticking out of the x " 0 plane, which makes the embedding dimension of f ´1p0q equal to three as it must be. This fact was stressed also in [Tei77, §3, section 3.5].

More material on the plane curve case is found in [G-L-S07, Chap. II, §2]. There are generalizations of these results to the cases where the fibers may be non reduced and have embedded components. There are definitions of the invariant δ which apply to this more general situation. We refer the reader to [L15] and the references therein.

General projections

For a reduced and equidimensional germ of complex analytic variety pX, 0q Ă pC N , 0q Whitney gave 6 possible definitions of tangent vectors ([Whi65]), the sets of which constitute tangent cones:

C 1 pX, 0q Ă C 2 pX, 0q Ă C 3 pX, 0q Ă C 4 pX, 0q Ă C 5 pX, 0q Ă C 6 pX, 0q,
and when the germ pX, 0q is smooth they all coincide with the tangent space T 0 X.

What is usually known as the tangent cone C X,0 is what Whitney defined as the cone C 3 pX, 0q and is constructed by taking limits of secants through the origin. This means that if we take a representative pX, 0q Ă pC N , 0q then a vector v P C N is in C 3 pX, 0q if there exists a sequence of points tp i u Ă Xzt0u tending to 0 and a sequence of complex numbers tλ i u Ă C ˚such that

λ i p i Ñ v.
Algebraically it is constructed by blowing up the point e 0 : Bl 0 X Ñ X and the fiber over the origin is the projectivized tangent cone e ´1 0 p0q " PC 3 pX, 0q. In particular it is a pure d-dimensional algebraic cone where d is the dimension of pX, 0q.

If pX, 0q is a curve then the cone C 3 pX, 0q is a finite number of lines, one for each branch of X. By abuse of language they are called the tangents to X at 0. Of course different branches may have the same tangent.

Definition 1.3.1 A linear projection π : pC N , 0q Ñ pC M , 0q with kernel D is called C 3 -general (with respect to X) if it is transversal to the tangent cone. That is D X C 3 pX, 0q " t0u.

Note that by the Weierstrass Preparation Theorem (see 1.1.8 and [De-P00, Thm 3.4.24]) the condition is equivalent to the fact that the map

π|C 3 pX, 0q : C 3 pX, 0q Ñ C 3 pC M , 0q " T C M ,0
is finite (proper with finite fibers), which implies M ě d. The restriction of a C 3 -general projection to X π|X : pX, 0q Ñ pC M , 0q satisfies π ´1p0q " t0u since otherwise the tangent cone to π ´1p0q, which is contained in C 3 pX, 0q, would be contained in D. Again by the Weierstrass Preparation Theorem, this is equivalent to π|X being a finite map. However the finiteness of π|X does not imply that the projection is C 3 -general; consider the projection of y 3 ´x2 " 0 to the x-axis.

Since C 3 pX, 0q is of dimension d, almost all (an open dense set of) linear projections π : pC N , 0q Ñ pC d`1 , 0q are C 3 -general for pX, 0q. Since we assume X to be equidimensional, this tells us that πpXq Ă C d`1 is a hypersurface. In the curve case (d=1) this guarantees the existence of linear projections with image a plane curve.

By [Chi89, Cor. 8.2] we have that C 3 pπpXq, 0q " π pC 3 pX, 0qq in C d`1 . We leave it as an exercise for the reader to verify that this last equality is an equality of Fitting images. Hint: use the specialization spaces X and Y to the tangent cones for X and C d`1 respectively (see [Gi-T18, §2, 2.4]) and the fact that the natural map X Ñ Y is finite by the Weierstrass Preparation Theorem because the genericity assumption is equivalent to the finiteness of the map C 3 pX, 0q Ñ T C d`1 ,0 , and apply [Mo-P89, Prop. 1.6]. Finally, a projection π : pC N , 0q Ñ pC d , 0q is C 3 -general for pX, 0q if and only if the map C 3 pX, 0q Ñ C 3 pC d , 0q " T C d ,0 which it induces is finite and surjective, and thus a ramified covering. These C 3 -general maps all induce on pX, 0q ramified analytic coverings pX, 0q Ñ pC d , 0q of degree equal to the multiplicity of pX, 0q.

The cone C 4 pX, 0q is constructed by taking limits of tangent vectors at smooth points. One can prove that it is equivalent to taking limits at 0 in the appropriate Grasmannian of tangent spaces at non singular points of X and so it is determined by the fiber over 0 of the Semple-Nash modification of a representative X of pX, 0q. Of course there is an analogous definition of a C 4 -general linear projection and they do have interesting equisingularity properties. However, since in the curve case the cones C 3 and C 4 coincide we will skip this part and ask the interested reader to look at [Chi89], [Stu72a] and [Stu72b].

The cone C 5 pX, 0q is constructed by taking limits of secants. This means that if we take a representative X Ă C N then a vector v P C N is in C 5 pX, 0q if there exist sequences of pairs of distinct points tp i u, tq i u Ă Xzt0u tending to 0 as i Ñ 8 and a sequence of complex numbers tλ i u Ă C ˚such that λ i pp i ´qi q Ñ v.

To prove that C 5 pX, 0q is an algebraic cone and have a bound for its dimension, take a small representative X Ă C n consider the (closed) diagonal embedding δ : X ãÑ X ˆX and blow up its image ∆:

e ∆ : Bl ∆ pX ˆXq Ñ X ˆX .
If we choose coordinates pz 1 , . . . , z N , w 1 , . . . , w N q of the ambient space C 2N , then we can obtain the space Bl ∆ pX ˆXq as the closure of the graph of the secant map defined away from the diagonal ∆ by: X ˆXz∆ ÝÑ P N ´1 pz, wq Þ ÝÑ rz 1 ´w1 : ¨¨¨: z N ´wN s.

So we have Bl ∆ pX ˆXq as a closed subspace of the product X ˆX ˆPN´1 , the map e ∆ is induced by the projection to X ˆX, and the exceptional fiber is the divisor D :" e ´1 ∆ p∆q Ă ∆ˆP N ´1 which comes with a map D Ñ ∆ such that for every point pq, qq P ∆ the fiber is the projective subvariety corresponding to the projectivization of the C 5 -cone of X at q, that is PC 5 pX, qq. This is roughly the way Whitney proved that the C 5 -cone is an algebraic variety in [Whi65, Th. 5.1]. Now C 5 pXq is the analytic space obtained by deprojectivization of the (fibers of) the divisor D and ψ corresponds to the pullback of e ∆ by δ:

X ˆCN Ą C 5 pXq / / ψ Bl ∆ pX ˆXq e ∆ X δ / / X ˆX
where the upper arrow is defined only outside of X ˆt0u. Note that the dimension of C 5 pXq is 2d, and the dimension of ψ ´1ppq " C 5 pX, pq for a smooth point p P X is equal to d since in this case we have C 5 pX, pq " T p X.

By the semicontinuity of the dimensions of the fibers of an analytic morphism, this implies that:

d ď dim C 5 pX, 0q ď 2d
Definition 1.3.2 A linear projection π : pC N , 0q Ñ pC M , 0q with kernel D is called generic (or C 5 -general) with respect to X if it is transversal to the cone C 5 pX, 0q. That is D X C 5 pX, 0q " t0u.

In other words, no limit at 0 of secants to X is contained in D.

Note that a generic projection is in particular C 3 -general and C 4 -general.

Proposition 1.3.3 Let pX, 0q Ă pC N , 0q be a reduced equidimensional germ of complex analytic variety of dimension d and π : pC N , 0q Ñ pC M , 0q a linear projection.

a) If π is generic then the restriction to X induces a homeomorphism with its image. b) pX, 0q is smooth if and only if dim C 5 pX, 0q " d Proof First of all note that the transversality to the cone C 5 pX, 0q implies that the restriction π|X is injective for a small enough representative of X. But then the induced map π|X : X Ñ C M is injective, continous and the map X Ñ πpXq is open since π is and so it should be a homeomorphism of X with its image πpXq. Now for bq: sufficiency is clear since pX, 0q smooth implies C 5 pX, 0q " T 0 X and so it is of dimension d. Conversely, if the dimension of C 5 pX, 0q is d there exist generic linear projections of X to C d . By aq this gives us a homeomorphism between pX, 0q and pC d , 0q. Note that π is also C 3 -general so it induces a ramified covering of degree equal to the multiplicity of pX, 0q, but the injectivity gives us multiplicity 1 and so pX, 0q is smooth.

For more on this and more general results see [Stu72a], [Stu77] and [Chi89, Section 9.4] An important thing to notice is that in the reducible case the cone C 5 pX, 0q contains but is not equal to the union of the C 5 -cones of its irreducible components. For instance if pX , 0q is a curve consisting of two smooth branches X 1 and X 2 then both cones C 5 pX i , 0q are one-dimensional but since pX, 0q is singular then by the previous result C 5 pX, 0q can not have dimension 1.

So now we have that if pX, 0q is singular then d `1 ď dim C 5 pX, 0q ď 2d, and for curves this gives dim C 5 pX, 0q " 2. This guarantees the existence of generic projections of curves to C 2 . Corollary 1.3.4 Let pX, 0q Ă pC N , 0q be a germ of reduced analytic curve. Then almost all (an open dense set of ) linear projections π : pC N , 0q Ñ pC 2 , 0q are generic and their Fitting images πpXq Ă C 2 are reduced plane curves homeomorphic to X. Moreover, π induces an analytic isomorphism Xzt0u Ñ πpXqzt0u.

Proof This follows from Proposition 1.2.6 and the fact that an analytic map C Ñ C which is a homeomorphism is an isomorphism.

1.3.1

The case of dimension 1.

In the case of curves we have the following important results:

Proposition 1.3.5 (see [B-G-G80, Prop IV.1])
Let pX, 0q Ă pC N , 0q be a germ of reduced analytic curve. If pX, 0q is singular then the cone C 5 pX, 0q is a finite union of 2-planes each one of them containing at least one tangent line to pX, 0q.

Proof We will only give an idea of the proof. By Proposition 1.3.3 the cone C 5 pX, 0q is two dimensional and by the blowup construction it has a finite number of irreducible components. So what one has to prove is that all the irreducible components are 2-planes. Again, by this blowup construction, any (direction of) line contained in C 5 pX, 0q can be picked off by lifting an arc pψ 1 , ψ 2 q : pC, 0q Ñ pX ˆX, p0, 0qq

to Bl ∆ pX ˆXq like pψ 1 ptq, ψ 2 ptq, rψ 1 ptq ´ψ2 ptqsq. Now each ψ i ptq is an arc pC, 0q Ñ pX, 0q and can be obtained using the parametrization of one of the branches of pX, 0q. Once you see this, what you have to do is consider the different cases and work out the calculations. The first case is when pX, 0q Ă pC N , 0q is irreducible of multiplicity n so according to subsection 1.0.2, in suitable coordinates we have a parametrization of the form:

ϕptq " ˜tn , ÿ iąn a 2,i t i , . . . , ÿ iąn a N,i t i
with the tangent line being the z 1 -axis r1 : 0 : ¨¨¨: 0s. For every n-th root of unity ω ‰ 1 the lifted arc t Þ Ñ pϕptq, ϕpωtq, rϕptq ´ϕpωtqsq P X ˆX ˆPN´1 will define a limit line ω P P N ´1 as t Ñ 0 which is distinct from the z 1 axis and if you define H ω as the 2-plane generated by the z 1 -axis and the line in C N corresponding to ω , then you can prove that

C 5 pX, 0q " H ω 1 Y . . . Y H ω n´1 ,
by verifying that any line obtained by lifting an arc is contained in one of these 2-planes. We note that they are not necessarily all different.

For the reducible case it is enough to consider two branches pX, 0q " pX 1 , 0q Y pX 2 , 0q. In this case you have that the C 5 -cone of each irreducible component pX i , 0q will be contained in C 5 pX, 0q but you will have additional components that come from the configuration of these two branches. For instance if they have different tangent lines 1 and 2 then all you have to add is the plane H 12 generated by these two lines.i.e.,

C 5 pX, 0q " C 5 pX 1 , 0q Y C 5 pX 2 , 0q Y H 12 .
When the two branches are tangent (have the same tangent line) then you have to play a game very similar to the irreducible case by reparametrizing your branches in such a way as to travel through them at the same "speed" and using roots of unity to find lines ω in the C 5 pX, 0q that are different from the tangent line and these will give you the additional 2-planes. i.e.,: -0 is not an isolated point of H X X.

C 5 pX, 0q " C 5 pX 1 , 0q Y C 5 pX 2 , 0q Y H ω 1 Y . . . Y H ω k . Proposition 1.3.6 (see [B-G-G80, Prop IV.2]) Let pX, 0q Ă pC N ,
-0 is an isolated point of H X X but the curve pπ H pXq, 0q is not reduced. -0 is an isolated point of H X X, the curve pπ H pXq, 0q is reduced but its Milnor number is greater than µ 0 .

Proof Let W 1 Ă GpN ´2, Nq be the open subset of the the Grassmannian of pN ´2q-planes of C N defined by the condition that H P W 1 if and only if 0 P C N is an isolated point of H X X. Let W Ă C 2N with coordinate system pa 1 , . . . , a N , b 1 , . . . , b N q be the associated open subset, where d " pa, bq P W if and only if the linear forms

a 1 z 1 `¨¨¨`a N z N and b 1 z 1 `¨¨¨`b N z N
are linearly independent and the N ´2 plane H d Ă C N they define is in W 1 . Let π d be the linear projection

π d : C N ÝÑ C 2 pz 1 , . . . , z N q Þ Ñ pa 1 z 1 `¨¨¨`a N z N , b 1 z 1 `¨¨¨`b N z N q
Note that for d P W the germ π d : pX, 0q Ñ pC 2 , 0q is finite, and if we denote by pπ d pXq, 0q Ă pC 2 , 0q the image germ with the Fitting structure then by [Mo-P89, Lemma 2.1] it is a (not necessarily reduced, but without embedded component, by Proposition 1.2.6) plane curve. We put all these projections in an analytic family by considering the map

Π : C N ˆW ÝÑ C 2 ˆW pz 1 , . . . , z N , dq Þ Ñ pπ d pz 1 , . . . , z N q, dq
Note that for every d P W the map germ Π : pX ˆW, p0, dqq Ñ `C2 ˆW, p0, dq ȋs finite. And since the analytic algebra O X ˆW,p0, dq is Cohen-Macaulay again by [Mo-P89, Lemma 2.1] we have a germ of hypersurface pΠpXˆ, W q, p0, dqq Ă `C2 ˆW, p0, dq ˘. By projecting to W Ă C 2N we obtain (by [G-L-S07, Thm B.8.11]) a flat map: G : pΠpX ˆW q, p0, dqq Ñ pW, dq.

Since the Fitting structure commutes with base change we have that the germ `G´1 pdq, p0, dq ˘is isomorphic to pπ d pXq, 0q, and so we have a flat deformation of pπ d pXq, 0q where all the fibers are plane curves.

Note that if ϕ : pC, 0q Ñ pC N , 0q, t Þ Ñ pϕ 1 ptq, . . . , ϕ N ptqq is the normalization of a branch of pX, 0q then the plane curve pπ d pXq, 0q is parametrized by:

t Þ Ñ pa 1 ϕ 1 ptq `¨¨¨`a N ϕ N ptq, b 1 ϕ 1 ptq `¨¨¨`b N ϕ N ptqq ,
and by varying d we get that the deformation space of G admits a parametrization in family.

Proof of a): When H d is transversal to C 5 pX, 0q then for every d 1 in a small neighborhood of d the pn ´2q-plane H d 1 is also transversal to C 5 pX, 0q and all the corresponding projections π d 1 are therefore generic. By Corollary 1.3.4 this tells us that π d 1 : Xzt0u Ñ G ´1pd 1 qzt0u is an analytic isomorphism for every d 1 sufficiently close to d. This implies:

• All the curves in the family G ´1pd 1 q have the same number of branches as X. and it contains (the direction of) k . Note that π d k p k q " q k ‰ 0 and so the plane curve pG ´1pd k q, q k q is singular which implies that µ ppπ d pXq, 0qq ą µ ppπ d k pXq, 0qq. Example 1.3.7 Let pX, 0q Ă pC 3 , 0q the germ of irreducible curve parametrized by t Þ Ñ pt 4 , t 5 , t 7 q then the tangent cone C 3 pX, 0q is the z 1 -axis. By taking other arcs t Þ Ñ pt 4 , ωt 5 , ω 3 t 7 q were ω P µ 4 zt1u and taking the limit as t Ñ 0 of the difference p0 : p1 ´ωqt 5 : p1 ´ω3 qt 7 q we get the z 2 ´axis as a limit of secants and we can deduce that the cone C 5 pX, 0q is the z 1 z 2 -plane. For d " p1, 0, 0, 0, 1, 0q the corresponding projection

π d pz 1 , z 2 , z 3 q " pz 1 , z 2 q
is C 5 -general and its image π d pX, 0q Ă pC 2 , 0q is the reduced plane curve y 4 ´x5 " 0 with Milnor number µ " 12.

On the other hand For d 0 " p1, 0, 0, 0, 0, 1q the corresponding projection π d 0 pz 1 , z 2 , z 3 q " pz 1 , z 3 q is not C 5 -general and its image π d 0 pX, 0q Ă pC 2 , 0q is the reduced plane curve y 4 ´x7 " 0 with Milnor number µ " 18. By taking d α " p1, 0, 0, 0, ´α2 , 1q we get a sequence of C 5 -general projections

π d α converging to π d 0 π d α pz 1 , z 2 , z 3 q " pz 1 , z 3 ´α2 z 2 q
Note that the plane curve X α :" π d α pXq has a singular point in pα 4 , 0q coming from the image of the secant going through the points pα 4 , α 5 , α 7 q and pα 4 , ´α5 , ´α7 q in X. Moreover as α tends to 0 these secants d α " r0 : 1 : α 2 s converge to the z 2 -axis r0 : 1 : 0s in P 2 which is precisely the intersection H d 0 X C 5 pX, 0q.

Main result

We have just seen that all (C 5 -)generic plane projections of a reduced analytic curve are equisingular. Now our objective is to prove that all equisingular germs of reduced plane curves are generic projections of a single space curve.

As we shall see, given a reduced plane curve pX, 0q Ă pC 2 , 0q this space curve corresponds to the one dimensional analytic algebra which is the Lipschitz saturation O s X,0 of O X,0 in the sense of [P-T69]. In doing so we will also give another reason why a) of Proposition 1.3.6 is true, since we shall see that a projection π is generic for a space curve pX, 0q Ă pC N , 0q if and only if it induces an isomorphism of the saturated algebras O s X,0 and O s πpX q,0 . In particular, two germs of reduced plane curves are equisingular (topologically equivalent) if and only if their saturations are analytically isomorphic. In order to define these saturations we need the theory of integral closure of ideals.

Integral closure of ideals

Our main references for this subsection are, , [Lip82], [Tei73] and .

Definition 1.4.1 Let I be an ideal in a ring R. An element r P R is said to be integral over I if there exists an integer h and elements a j P I j , j " 1, . . . , h, such that r h `a1 r h´1 `a2 r h´2 `¨¨¨`a h´1 r `ah " 0.

The set of all elements of R that are integral over I is an ideal called the integral closure of I and denoted by I. We say that I is integrally closed if I " I. If I Ă J are ideals we say that J is integral over I if J Ă I.

Remark 1.4.2 The following properties are easily verified:

1. I Ă I. For each r P I choose n " 1 and a 1 " ´r.

2. If I Ă J are ideals then I Ă J since an integral dependence equation for r over I is also an an integral dependence equation for r over J.

I Ă ?

I since the integral dependence equation implies r n P xa 1 , . . . , a n y Ă I. 4. Radical ideals are integrally closed. 5. If ϕ : R Ñ S is a ring morphism and I Ă S is an integrally closed ideal of S then ϕ ´1pI q is an integrally closed ideal of R.

A related concept is that of reduction: For ideals J Ă I Ă R we say that J is a reduction of I if there exists a non-negative integer n such that I n`1 " J I n . This implies that I " J. We can express integral dependence using equalities of ideals and modules.

Proposition 1.4.3 (see Chapter 1], Prop 1.1.7,Cor. 1.1.8 & Cor. 1.2.2]) For any element r P R and ideal I Ă R. The following are equivalent: a) r P I. b) There exists an integer k such that pI `rq k " I pI `rq k´1 . c) I is a reduction of I`ă r ą. d) There exists a finitely generated R´module M such that r M Ă I M and if there exists a P R such that aM " 0, then there exists an integer such that ar " 0. .

A very important corollary of this Proposition is that I Ă R is an integrally closed ideal of R and you can find a complete proof of this fact in Cor. 1.3.1].

We have that I Ă I Ă ? I, but in fact the integral closure is much "closer" to I than to the radical and a very good family of examples in which it is easy to calculate and compare is that of monomial ideals in Ctz 1 , . . . , z d u, which are the ideals generated by monomials. We begin with an example:

Example 1.4.4 For the ideal I " xx 4 , x y 2 , y 3 yCtx, yu we have that I " xx 4 , x 3 y, x y 2 , y 3 y and ? I " xx, yy.

The exponent set of I consists of all integer lattice points in the yellow region below:

exp.x exp.y (4,0)

(0,3) (1,2) (3,1)
x 4 y 3

x y 2

x 3 y Fig. 1.1 The point p3, 1q representing the monomial x 3 y is in the convex hull of the yellow region, whose integral points represent monomials in I . The integral dependence relation is px 3 yq 2 ´x5 .x y 2 " 0.

Similarly, in Ctz 1 , . . . , z d u we have xz n 1 , . . . , z n d y " xz 1 , . . . , z d y n . The exponent vector of a monomial m " z m 1 1 ¨¨¨z m d d is pm 1 , . . . , m d q P N d . For any monomial ideal I, the set of all exponent vectors of all the monomials in I is called the exponent set of I. Since a monomial m is in I if and only if it is a multiple in Ctz 1 , . . . , z d u of one of the monomial generators of I, the exponent set of I consists of all those points of N d which are componentwise greater or equal than the exponent vector of one of the monomial generators of I. Moreover one can prove that I is monomial and its exponent set is equal to all the integer lattice points in the convex hull of the set of exponents of elements of I. (See [Tei04, §3, §4], [Tei82, Chap.1, §2], Props 1.4.2 & 1.4.6]).

To understand how this theory can be used in the setting of complex analytic geometry the following result is fundamental.

Theorem 1.4.5 ([Lej-T08, Thm 2.1, p. 799]) Let X be a reduced complex analytic space. Let Y Ă X be a closed, nowhere dense, analytic subspace of X, and x a point in Y . Let I Ă O X be the coherent ideal defining Y , and let J Ă O X be another coherent ideal. Let I (resp. J) be the stalk of I (resp. J ) at x. Then the following statements are equivalent:

1. J Ă I, 2. For every germ of morphism φ : pC, 0q Ñ pX, xq φ ˚J ¨OC,0 Ă φ ˚I ¨OC,0 , 3. For every morphism π : X 1 Ñ X such that X 1 is a normal analytic space, π is proper and surjective, and I ¨OX 1 is locally invertible, there exists an open subset U Ă X containing x, such that:

J ¨OX 1 |π ´1pU q Ď I ¨OX 1 |π ´1pU q, 3 ˚. If Π : Bl I X Ñ X denotes the normalized blowup of X along I, then there exists an open subset U Ă X containing x, such that:

J ¨OBl I X |Π ´1pU q Ď I ¨OBl I X |Π ´1pU q,
4. Let V Ă X be a neighborhood of x, where both J and I are generated by their global sections. Then for every system of generators g 1 , . . . , g m of ΓpV, Iq and every f P ΓpV, J q, there exist a neighborhood V 1 of x in V and a constant C such that:

| f pyq| ď C sup i"1, ..., m |g i pyq| for every y P V 1 .
Let us take a closer look at statement 2: For any arc ϕ : pC, 0q Ñ pX, 0q Ă pC N , 0q we have a corresponding morphism of analytic algebras

ϕ ˚: O X,0 " Ctz 1 , . . . , z N u{a ÝÑ Cttu z i `a Þ Ñ ϕ i ptq " t m i u i ptq
where m i ě 1 and u i ptq is a unit in Cttu. So if I Ă O X,0 is an ideal then ϕ ˚pI qO C,0 " xt k yCttu for some integer k and an element g P O X,0 is in I if and only if for any such arc ϕptq the order of the series g pϕ 1 ptq, . . . , ϕ N ptqq is greater or equal than this k. The fact that the normalized blowing-up map is proper implies that the condition of statement 2 needs to be verified only for finitely many arcs. Since the general statement is somewhat cumbersome, let us illustrate how this works in the case where the ideal I a complete intersection defining the origin in pX, 0q. Let I " xh 1 , . . . , h d y Ă O X,0 . The blowing up Bl I X of I in X is the subspace of X ˆPd´1 defined by the d´1 equations h 1 T 1 " h 2 T 2 " ¨¨¨" h d T d , again a complete intersection. The fiber of the natural projection Bl I X Ñ P d´1 over a point t P P d´1 with coordinates pt 1 : t 2 : ¨¨¨: t d q is a curve in Bl I X which is isomorphic to its image in X defined by the equations h i t j ´hj t i " 0. So we can view Bl I X as a family of curves C t on X parametrized by P d´1 , which is the exceptional divisor of the map Bl I X Ñ X. When we pass to the normalization n : Bl I X Ñ Bl I X, by general Theorems on normalization (see Proposition 1.2.9 and use the fact that by generic flatness there is a dense open U Ă P d´1 where δ is constant), there exists a Zariski dense open subset U Ă P d´1 such that n ´1pU q is a non singular divisor in a non singular space n ´1ppX ˆUq X Bl I Xq, and for each point t P U the map n induces a normalisation of the curve C t . This normalization is then a union of disks, one for each irreducible component of C t , and each disk transversal to n ´1pP d´1 q in n ´1ppX ˆUq X Bl I Xq. Because a meromorphic function on a normal space is holomorphic if it has no poles in codimension one, to verify that an element g P O X,0 is in I, it suffices to verify that for some t P U, the order of vanishing of g along each arc parametrizing a branch of C t is larger than the order of vanishing of the ideal I. Because of what we have just seen, the order of vanishing along these arcs will, after lifting to Bl I X, translate as the order of vanishing along some irreducible component of the exceptional divisor in Bl I X. Since the ideal I is locally principal on Bl I X, to prove that g P I it suffices to prove that after lifting to Bl I X the function g becomes a multiple of the local equations of the exceptional divisor. But the polar set of the quotient of g by that equation is contained in that exceptional divisor and the inequalities of orders imply that there are no poles at a general point of each irreducible component. Because Bl I X is normal, there are no poles anywhere and on Bl I X the pull back of the function g is indeed in the pull back of the ideal I so that g is in I.

We shall use this below to describe the saturation.

With this at hand we can now characterize C 3 -general projections in terms of integral closure of ideals. Let pX, 0q Ă pC N , 0q be a reduced germ of analytic space of pure dimension d. Let us choose coordinates z 1 , . . . , z N on C N , denote by L the linear subspace of C N defined by z 1 " ¨¨¨" z d " 0 and let a be the ideal of O X,0 generated by the images of z 1 , . . . , z d .

Proposition 1.4.6 The restriction to pX, 0q π|pX, 0q : pX, 0q Ñ pC d , 0q pz 1 , . . . , z N q Þ Ñ pz 1 , . . . , z d q of the linear projection π with kernel L is C 3 -general if and only if a " m where m " xz 1 , . . . , z N yO X,0 is the maximal ideal of the analytic algebra O X,0 .

Proof Recall that π is C 3 -general if and only if C 3 pX, 0q X L " t0u. Let " ra 1 : ¨¨¨: a N s P P N ´1 be a line in the (projectivized) tangent cone C 3 pX, 0q, then Ć L if and only if a i ‰ 0 for some i P t1, . . . , du. Note that any arc ϕ : pC, 0q Ñ pX, 0q determines a line in C 3 pX, 0q, the limit as t Ñ 0 of t ÝÑ rϕ 1 ptq : ¨¨¨: ϕ N ptqs P P N ´1, and conversely any line in the tangent cone can be obtained through an arc since it corresponds to a point in the fiber over 0 of the blowing-up Bl 0 X Ñ X. On the other hand, for every arc ϕ : pC, 0q Ñ pX, 0q we have that

ϕ ˚paqO C,0 " xϕ 1 ptq, . . . , ϕ d ptqyCttu " xt k yCttu,
where k " mintord 0 ϕ i ptq | i " 1, . . . , du. Finally a i ‰ 0 for some i P t1, . . . , du if and only if for all j P td `1, . . . , Nu ord 0 ϕ j ptq ě k " mintord 0 ϕ i ptq | i " 1, . . . , du if and only if ϕ ˚pz j q P ϕ ˚paqO C,0 if and only if z j P a for all j P td `1, . . . , Nu, that is, a " m. By a linear change of coordinates in C N we can always place ourselves in the setting of the previous result. But the theory of integral closure also gives us an algebraic way to prove that for a given germ pX, 0q of pure dimension d almost all linear projections π : pC N , 0q Ñ pC d , 0q are C 3 -general as stated in the following result (For a proof see [Mat89,Thm 14.14]) Theorem 1.4.7 (Rees-Samuel) Let O X,0 be a d-dimensional analytic algebra with maximal ideal m " xz 1 , . . . , z N y. Then if y i " ř N j"1 λ i j z j for 1 ď i ď d are d "sufficiently general" C-linear combinations of z 1 , . . . , z n the ideal a " xy 1 , . . . , y d y satisfies a " m.

We can take this one step further by considering another important aspect of this theory, namely its relation with multiplicity. For a local Noetherian ring pR, mq and an m´primary ideal a Ă R we can define a Hilbert Samuel function

k P N Þ Ñ dim R{m R{a k .
The result is that for large enough k the Hilbert-Samuel function behaves like a polynomial of degree equal to the dimension of R and its leading coefficient is of the form epaqk d {d!, where epaq is a positive integer called the multiplicity of the ideal a. In the case R is the analytic algebra O X,0 of a germ pX, 0q and a " m it IS the multiplicity of the germ. (See [De-P00, Section 4.2])

Theorem 1.4.8 (Rees)(see [Ree61,Thm 3.2], Thm 11.3.1]) Let pO X,0 , mq be a reduced and equidimensional analytic algebra and a Ă b two m´primary ideals. Then a " b if and only if epaq " epbq.

A geometric interpretation of this result is described by Lipman in [Lip82]. Let pX, 0q be a germ of reduced and equidimensional singularity of dimension d with associated analytic algebra pO X,0 , mq. Every m´primary ideal is generated by at least d elements, and every d´tuple p f 1 , . . . , f d q of elements of m defines a map-germ F : pX, 0q Ñ pC d , 0q. Now, the ideal a " x f 1 , . . . , f d y is m´primary if and only if F is finite. As we have mentioned before you can prove that such an F : pX, 0q Ñ pC d , 0q is then a ramified analytic cover of degree equal to epaq and by Rees' Theorem this degree will be the multiplicity of pX, 0q (" epmqq if and only if a " m.

Moreover using Nakayama's Lemma one checks that a is a reduction of m (equivalently a " m) if and only if in the graded C´algebra

gr m O " à kě0 m k {m k`1 , with m 0 " O
(which is the homogeneous coordinate ring of the projectivized tangent cone PC 3 pX, 0q see [Gi-T18, Section 2.4]) the images f i of the f i in m{m 2 generate an irrelevant ideal (that is, an ideal containing all elements of gr m O of sufficiently large degree so that its zero locus in projective space is empty).

What this last condition means is that first of all the f i are linearly independent over C, so that there is an embedding of the germ pX, 0q into pC N , 0q for some N and a linear projection π : pC N , 0q Ñ pC d , 0q such that its restriction to pX, 0q is germwise the F associated above to p f 1 , . . . , f d q and secondly, since a " m by Proposition 1.4.6 the projection π is C 3 -general.

We end this section by establishing a result analogous to Proposition 1.4.6 but with respect to generic projections of curves.

Definition 1.4.9 Let ϕ 1 : R Ñ A 1 and ϕ 2 : R Ñ A 2 be morphisms of Canalytic algebras. There is a unique C-analytic algebra, denoted A 1 b R A 2 , together with morphisms θ i :

A i Ñ A 1 b R A 2 , i " 1, 2, such that θ 1 ˝ϕ1 " θ 2 ˝ϕ2
and for every pair of morphisms of C-analytic algebras ψ

1 : A 1 Ñ B, ψ 2 : A 2 Ñ B satisfying ψ 1 ˝ϕ1 " ψ 2 ˝ϕ2 there is a unique morphism of C-analytic algebras ψ : A 1 b R A 2 Ñ B making the whole diagram commute. The algebra A 1 b R A 2 is called the analytic tensor product of A 1 and A 2 over R. A 1 θ 1 # # ψ 1 R ϕ 1 ? ? ϕ 2 A 1 b R A 2 ψ / / B A 2 θ 2 ; ; ψ 2 E E
Geometrically this analytic tensor product is the operation on the analytic algebras that corresponds to the fibre product of analytic spaces. Given holomorphic maps φ 1 : pX 1 , p 1 q Ñ pY, qq and φ 2 : pX 2 , p 2 q Ñ pY, qq we have the fibre product: 1. When R " C in the definition, the analytic tensor product

X 1 ˆY X 2 Π 2 Π 1 / / X 1 φ 1 X 2 φ 2 / / Y which induces the corresponding diagram of analytic algebras O Y,q ϕ 2 ϕ 1 / / O X 1 , p 1 O X 2 , p 2 / / O X 1 ˆY X 2 ,pp 1 , p 2 q that is, the analytic algebra O X 1 ˆY X 2 ,pp 1 , p 2 q is isomorphic to O X 1 , p 1 b O Y, q O X 2 , p 2 .
O X 1 , p 1 b C O X 2 , p 2
is the analytic algebra corresponding to the product germ pX 1 ˆX2 , pp 1 , p 2 qq. Moreover if O X 1 , p 1 " Ctzu{I and O X 2 , p 2 " Ctwu{J with z " pz 1 , . . . , z N q and w " pw 1 , . . . , w M q then

O X 1 , p 1 b C O X 2 , p 2 " Ctz, wu xICtz, wu `JCtz, wuy .
2. In general if pX 1 , p 1 q Ă pC N , 0q, pX 2 , p 2 q Ă pC M , 0q and pY, qq Ă pC k , 0q with O Y,q " Cty 1 , . . . , y k u{K, denoting by y k pzq (resp. y k pwq a representative in

Ctzu (resp. Ctwu) of the image of y k in O X 1 , p 1 (resp. O X 2 , p 2 by the structure maps O Y,q Ñ O X i , p i i " 1, 2, then O X 1 , p 1 b O Y, q O X 2 , p 2 " Ctzu I b O Y, q
Ctwu J -Ctz, wu ICtz, wu `JCtz, wu `xy 1 pzq ´y1 pwq, . . . , y k pzq ´yk pwqyCtz, wu .

Let pX, 0q Ă pC N , 0q be a germ of reduced singular curve. By Proposition 1.3.5 the C 5 cone is a finite union of 2-planes of

C n C 5 pX, 0q " H 1 Y . . . Y H r .
If we let π : pC N , 0q Ñ pC 2 , 0q denote the linear projection to the first two coordinates pz 1 , . . . , z N q Þ Ñ pz 1 , z 2 q then π is generic if and only if πpH i q " C 2 for i " 1, . . . , r. Recall that the construction of C 5 pX, 0q goes through the blowup of the diagonal of X ˆX, so let I ∆ Ă O X ˆX,p0,0q be the ideal defining this diagonal

I ∆ " xz 1 ´w1 , . . . , z N ´wN yO X ˆX,p0,0q . Proposition 1.4.11 Let I ∆ 2 Ă O X ˆX,p0
,0q be the ideal coming from the projection π, that is,

I ∆ 2 " xz 1 ´w1 , z 2 ´w2 yO X ˆX,p0,0q . Then π is generic if and only if I ∆ 2 " I ∆ .
Proof The proof is now very similar to the C 3 -general case, and since I ∆ 2 Ă I ∆ all we have to prove is that genericity is equivalent to the inclusion

I ∆ Ă I ∆ 2 .
Let L " V pz 1 , z 2 q be the kernel of π. Then π is generic if and only if C 5 pX, 0q X L " t0u. Let " ra 1 : ¨¨¨: a N s P P N ´1 be a line in the (projectivized) cone C 5 pX, 0q, then Ć L if and only if a i ‰ 0 for some i P t1, 2u. This time the lines in C 5 pX, 0q are determined by taking the limit as t Ñ 0 of the secants associated to pairs of arcs pϕ, ψq : pC, 0q Ñ pX ˆX, p0, 0qq t ÝÑ rϕ 1 ptq ´ψ1 ptq : ¨¨¨: ϕ N ptq ´ψN ptqs P P N ´1 Again for every such pair of arcs pϕ, ψq : pC, 0q Ñ pX ˆX, p0, 0qq we have that pϕ, ψq ˚pI ∆ 2 qO C,0 " xϕ 1 ptq ´ψ1 ptq, ϕ 2 ptq ´ψ2 ptqyCttu " xt k yCttu

where k " mintord 0 pϕ 1 ptq ´ψ1 ptqq, ord 0 pϕ 2 ptq ´ψ2 ptqqu. Finally a i ‰ 0 for some i P t1, 2u if and only if for all j P t3, . . . , Nu ord 0 pϕ j ptq ´ψj ptqq ě k " mintord 0 pϕ 1 ptq ´ψ1 ptqq, ord 0 pϕ 2 ptq ´ψ2 ptqqu if and only if pϕ, ψq ˚pz j ´wj q P pϕ, ψq ˚pI ∆ 2 qO C,0 if and only if z j ´wj P I ∆ 2 for all j P t3, . . . , Nu, that is, I ∆ Ă I ∆ 2 . ˝

Lipschitz Saturation

Let n ˚: O X,0 ãÑ O X,0 be the integral closure of a reduced complex analytic algebra which is a quotient of Ctz 1 , . . . , z N u. Recall that O X,0 is a direct sum of normal analytic algebras (in particular integral domains), one for each irreducible component of the germ pX, 0q. By Definition 1.4.9 the following commutative diagram determines a unique morphism Ψ of direct sums of analytic algebras:

O X,0 θ 1 % % ψ 1 C 2 2 , , / / O X,0 = = ! ! O X,0 b C O X,0 Ψ / / O X,0 b O X,0 O X,0 O X,0 θ 2 9 9 ψ 2 C C where θ 1 p f q " f b C 1 and θ 2 p f q " 1 b C f . Note that the map Ψ : O X,0 b C O X,0 Ñ O X,0 b O X,0 O X,0
is the morphism of sums of analytic algebras corresponding to the inclusion ´X ˆX X, p0, 0q ¯ãÑ ´X ˆX, p0, 0q ¯. By remark 1.4.10 if we denote by n : pX, 0q Ñ pX, 0q the normalization map and O X,0 " À r i"1 Ctt i u{J i pt i q with t i " pt i,1 , . . . , t i, m i q, then

O X,0 b C O X,0 " à i, j
Ctt i , u j u xJ i pt i q, J j pu j qy and Ψ is a surjection with kernel

I ∆ " xz 1 b C 1 ´1 b C z 1 , . . . , z N b C 1 ´1 b C z N yO X,0 b C O X,0 .
Definition 1.4.12 Let I ∆ be the kernel of the morphism Ψ above. We define the Lipschitz saturation O s X,0 of O X,0 as the algebra O s X,0 :"

! f P O X,0 | θ 1 p f q ´θ2 p f q P I ∆ ) " ! f P O X,0 | f b C 1 ´1 b C f P I ∆ ) .
Example 1.4.13 Let pX, 0q Ă pC 3 , 0q be the irreducible curve with normalization map:

η : pC, 0q ÝÑ pX, 0q t Þ Ñ pt 4 , t 6 , t 7 q
In this setting the map Ψ above is

Ψ : O X,0 b C O X,0 Ñ O X,0 b O X,0 O X,0 Ψ : Ctt, uu ÝÑ Ctt, uu
xt 4 ´u4 , t 6 ´u6 , t 7 ´u7 y .

The maps θ i are just inclusions, Cttu ãÑ Ctt, uu, Ctuu ãÑ Ctt, uu and the ideal

I ∆ " xt 4 ´u4 , t 6 ´u6 , t 7 ´u7 y. By definition O s X,0 :" ! f P Cttu | f ptq ´f puq P I ∆ )
and note that O X,0 Ă O s X,0 . For example t 5 P Cttu is in O s X,0 if and only if t 5 ´u5 is in I ∆ . By taking the arc φpτq " pτ, ´τq we have that φ ˚I∆ O C,0 " xτ 7 y and φ ˚pt ´u5 q " 2τ 5 R φ ˚I∆ O C,0 , so by Theorem 1.4.5-2 the element t 5 P Cttu " O X,0 is not in the Lipschitz saturation O s X,0 . For this particular arc we have φ ˚pt 9 ´u9 q " 2τ 9 P φ ˚I∆ O C,0 , and one can actually prove that t 9 P O s X,0 . As we shall see later on, in fact O s X,0 " Ctt 4 , t 6 , t 7 , t 9 u.

We are going to show that the Lipschitz saturation O s X,0 is always an analytic algebra, even if the germ pX, 0q is not irreducible. To begin to understand why this is true, let's look at the irreducible case. Define the map

α : O X,0 Ñ O X,0 b C O X,0 f Þ Ñ θ 1 p f q ´θ2 p f q " f pzq ´f pwq
It is not a ring map, however if n ˚: O X,0 ãÑ O X,0 denotes the inclusion coming from the normalization map n : X Ñ X then αpn ˚pO X,0 qq " αpn ˚pm X,0 qq and I " Ker Ψ " xαpn ˚pm X,0 qqy. By Definition 1.4.9 O X,0 b O X,0 O X,0 is an O X,0 -algebra, in particular an O X,0 -module. However, an interesting point is that since n : X Ñ X is a finite map, by Lemma 1.89] this algebra is isomorphic to the algebraic tensor product O X,0 b O X,0 O X,0 , so for instance

Ctt, uu xt 2 ´u2 , t 3 ´u3 y -Cttu b Ctt 2 , t 3 u Ctuu Lemma 1.4.14 The map Ψ ˝α : O X,0 ÝÑ O X,0 b O X,0 O X,0
is a morphism of O X,0 -modules.

Proof Indeed for r P O X,0 and f P O X,0 :

r f α Þ Ñ rpzq f pzq ´rpwq f pwq Ψ Þ Ñ rpzq f pzq ´rpwq f pwq `I
but rpzq " rpwq modpIq so rpzq f pzq´rpwq f pwq " prpzq`Iqp f pzq´f pwq`Iq " rpΨ ˝αqp f q.

By definition I ∆ is an ideal of O X,0 b C O X,0 and since Ψ is a surjective ring homomorphism we have that ΨpI ∆ q Ă O X,0 b O X,0 O X,0 is an ideal, in particular it is an O X,0 -module. But this implies that

pΨ ˝αq ´1 pΨpI ∆ qq " α ´1pI ∆ q " O s X,0 Ă O X,0
is an O X,0 -module. Lemma

1.4.15 The Lipschitz saturation O s X,0 is an O X,0 -algebra and a direct sum of analytic algebras.

Proof Since O s X,0 is a submodule of the Noetherian module O X,0 , it is a finitely generated O X,0 -module. Even more, you can easily check that O s X,0 is closed under multiplication, so it is an O X,0 -algebra and by Cor. 3.3.25 & 3.3.26] this implies that O s X,0 is a direct sum of analytic algebras. Indeed, take f 1 , f 2 P O s X,0 then pΨ ˝αqp f 1 q " f 1 pzq ´f1 pwq `I∆ P ΨpI ∆ q, but it is an ideal so p f 2 pzq `I∆ qp f 1 pzq ´f1 pwq `I∆ q P ΨpI ∆ q. Analogously p f 1 pwq `I∆ qp f 2 pzq ´f2 pwq `I∆ q P ΨpI ∆ q by taking their sum we get that pΨ ˝αqp f 1 f 2 q " f 1 pzq f 2 pzq ´f1 pwq f 2 pwq `I∆ P ΨpI ∆ q which implies that f 1 f 2 P O s X,0 as claimed. Before proving that O s X,0 is actually an analytic algebra we would like to give an idea of how things work in the non-irreducible case so suppose there are two irreducible components pX, 0q " pX 1 , 0q Y pX 2 , 0q. As we said before X is then a multigerm pX 1 , pq \ pX 2 , qq and

O X,0 " O X 1 ,0 À O X 2 ,0 " O X 1 , p À O X 2 ,q .
Since the analytic tensor product should be the algebraic counterpart of the fibre product then we should consider/define

O X,0 b O X,0 O X,0 " O X 1 , p b O X,0 O X 1 , p 'O X 1 , p b O X,0 O X 2 ,q 'O X 2 ,q b O X,0 O X 1 , p 'O X 2 ,q b O X,0 O X 2 ,q
and analogously for O X,0 b C O X,0 . By componentwise taking the ring maps Ψ i j coming from the universal property of the irreducible case, for example:

Ψ 12 : O X 1 , p b C O X 2 ,q Ñ O X 1 , p b O X,0 O X 2 ,q
we get the ring map Ψ as before with kernel I ∆ " I 11 ' I 12 ' I 21 ' I 22 . The map α should now be defined as

α : O X,0 ÝÑ O X,0 b C O X,0 p f 1 , f 2 q Þ Ñ
p f 1 pzq ´f1 pwq, f 1 pzq ´f2 pwq, f 2 pzq ´f1 pwq, f 2 pzq ´f2 pwqq and we get the same definition for the Lipschitz saturation O s X,0 :"

! f " p f 1 , f 2 q P O X,0 | αp f q P I ∆
) .

More importantly both Lemmas remain valid. Note that in this context of two irreducible components we have αp f q P I ∆ if and only if f 1 pzq ´f1 pwq P I 11 , f 1 pzq ´f2 pwq P I 12 , f 2 pzq ´f1 pwq P I 21 and f 2 pzq ´f2 pwq P I 22 .

Proposition 1.4.16 (See [P-T69, Theorem 1.2], [Tei82, Prop. 6.1.1]) The algebra O s X,0 is the ring of germs of meromorphic functions on pX, 0q which are locally Lipschitz with respect to the ambient metric.

Proof Recall that O X,0 is the ring of meromorphic functions on pX, 0q that are locally bounded and a Lipschitz meromorphic function is locally bounded. Now if h P O s X,0 we need to prove that there exists a real positive constant C ą 0 such that for every couple px 1 , x 2 q P XzSing X ˆXzSing X (in a small enough representative) we have

|hpx 1 q ´hpx 2 q| ď C||x 1 ´x2 ||.
Let n : X Ñ X Ă C n be the normalization map. In the irreducible case where O X,0 " Ctz 1 , . . . , z N u{x f 1 , . . . , f s y and O X,0 " Ctt 1 , . . . , t m u{Jptq, the map n induces a morphism of analytic algebras which may be described by n ˚: O X,0 ÝÑ O X,0 z i Þ Ñ z i pt 1 , . . . , t m q " z i ptq and refering to the maps α and Ψ as above we have that I ∆ " Ker Ψ " xz 1 ptq ´z1 puq, . . . , z N ptq ´zN puqy.

By definition, h P O s X,0 if αphq " hptq ´hpuq P I ∆ and by Theorem 1.4.5 there exists a constant C such that |hptq ´hpuq| ď C sup |z i ptq ´zi puq| " C||zptq ´zpuq||, with pzptq, zpuqq P X ˆX and so h is Lipschitz. Reading the proof in the opposite sense gives that a meromorphic, locally Lipschitz function h is necessarily in O s X,0 . If pX, 0q has r irreducible components then X is a multigerm and then we have r maps n k : pX k , x k q Ñ pX k , 0q Ă pX, 0q with coordinate functions z 1 pt k q, . . . , z N pt k q. Then for h " ph 1 , . . . , h r q P O X,0 we have that αphq " ´hi pt i q b1 ´1 bh j pu j q ¯i, j P À i, j"r i, j"1 O X i ,0 b C O X j ,0 , and

I ∆ "
i, j"r à i, j"1 I i j with I i j " xz 1 pt i q b1´1 bz 1 pu j q, . . . , z N pt i q b1´1 bz N pu j qyO Xi,0 b C O X j ,0

and αphq P I ∆ if and only if h i pt i q b1 ´1 bh j pu j q P I i j for all pi, jq. So in the spirit of Example 1.1.7 the "coordinate" h i of h indicates you how to evaluate h in points of the corresponding irreducible component pX i , 0q of pX, 0q and for i ‰ j the condition h i pt i q b1 ´1 bh j pu j q P I i j tells you that the Lipschitz condition must also be satisfied when you take points in different irreducible components. Corollary 1.4.17 (See [P-T69, Corollary 1.3]) Let pX, 0q Ă pC N , 0q be a reduced germ of complex analytic singularity. The ring O s X,0 is an analytic algebra.

Proof We already proved in Lemma 1.4.15 that O s X,0 is a direct sum of analytic algebras, but if there were more than one, the function p1, 0, . . . , 0q P O s X,0 would not be Lipschitz, contradicting Proposition 1.4.16. From

Lemma 1.4.15 we have injective ring morphisms

O X,0 ãÑ O s X,0 ãÑ O X,0 .
Since O X,0 is contained in the total ring of fractions QpO X,0 q, the total ring of fractions of the Lipschitz saturation O s X,0 coincides with QpO X,0 q and by transitivity of integral dependence the normalizations also coincides i.e., O s X,0 " O X,0 . In terms of holomorphic maps we have:

X n s ÝÑ X s ζ ÝÑ X,
where X s is the germ of complex analytic singularity corresponding to the analytic algebra O s X,0 , the map n s : X Ñ X s is the normalization map of X s , ζ : pX s , 0q Ñ pX, 0q is finite and induces an isomorphism outside the non-normal locus of X, and n " ζ ˝ns : X Ñ X is the normalization map of X.

Definition 1.4.18 The germ pX s , 0q together with the finite map ζ : pX s , 0q Ñ pX, 0q is called the Lipschitz saturation of pX, 0q.

Lemma 1.4.19 Let pX, 0q Ă pC n , 0q be a reduced germ of complex analytic singularity, then ´Os X,0 ¯s " O s X,0 .

Proof Following the notation of Lemma 1.4.14 we have the maps:

O X,0 ãÑ O s X, o ãÑ O X,0 α ÝÑ O X,0 b C O X,0 ,
and this induces a map

O X,0 b O X,0 O X,0 - O X,0 b C O X,0 xαpO X,0 qy ÝÑ O X,0 b C O X,0 xαpO s X,0 qy -O X,0 b O s X,0 O X,0 that makes the following diagram commute O X,0 b C O X,0 Ψ s ( ( Ψ O X,0 b O X,0 O X,0 / / O X,0 b O s X,0 O X,0
If we denote I ∆ " Ker Ψ and I ∆ s " Ker Ψ s we have I ∆ Ă I ∆ s . Now by definition we have O s X,0 " th P O X,0 | αphq P I ∆ u so αpO s X,0 q Ă I ∆ which implies I ∆ " I ∆ s and so ´Os X,0 ¯s " th P O X,0

| αphq P I ∆ s " I ∆ u " O s X,0 . 1.4.3
The case of dimension 1

Let pX, 0q Ă pC 2 , 0q be a germ of reduced plane curve, and ζ : pX s , 0q Ñ pX, 0q Ă pC 2 , 0q the finite map given by the Lipschitz saturation of pX, 0q. What we want to emphasize is that this map can always be realized as a linear projection on suitable representatives. Indeed, any representative of pX s , 0q Ă pC m , 0q can be re-embedded as the graph of ζ in C m`2 , namely by the map X s Ñ C 2 ˆCm : p Þ Ñ pζ 1 ppq, ζ 2 ppq, pq. The map ζ is now the projection of pX s , 0q to pX, 0q by the first two coordinates: pz 1 , . . . , z m`2 q Þ Ñ pz 1 , z 2 q.

Proposition 1.4.20 (See [Tei82, Proposition 6.2.1]) For a germ of reduced plane curve pX, 0q Ă pC 2 , 0q the Lipschitz saturation map ζ : pX s , 0q Ñ pX, 0q is a generic projection.

Proof Suppose first that pX, 0q is irreducible, in this case we have the holomorphic maps pC, 0q

η s ÝÑ pX s , 0q Ă pC m`2 , 0q ζ ÝÑ pX, 0q Ă pC 2 , 0q t Þ Ñ pz 1 ptq, z 2 ptq, z 3 ptq, . . . , z m`2 ptqq Þ Ñ pz 1 ptq, z 2 ptqq
By Proposition 1.4.11 we have to prove that the ideals I ∆ s " xz 1 ´w1 , . . . , z m`2 ẃm`2 y and I ∆ s 2 " xz 1 ´w1 , z 2 ´w2 y have the same integral closure in O X s ˆX s ,p0,0q . In this coordinate system we have the normalization map η s : O X s ,0 ãÑ O X,0 given by Ctz

1 , . . . , z m`2 u I ãÝÑ Cttu z i Þ Ñ z i ptq i " 1, 2 ; z j`2 Þ Ñ z j`2 ptq j " 1, . . . m, which induces the morphism θ : O X s ˆX s ,p0,0q " Ctz 1 , . . . , z m`2 , w 1 , . . . , w m`2 u Ipzq `Ipwq ãÝÑ Ctt, uu " O X,0 b C O X,0 z i Þ Ñ z i ptq i " 1, 2 ; z j`2 Þ Ñ z j`2 ptq j " 1, . . . m w i Þ Ñ z i puq i " 1, 2 ; w j`2 Þ Ñ z j`2 puq j " 1, . . . m.
But from the proof of Lemma 1.4.19 we have that the ideals I ∆ s 2 " xz 1 ptq ź1 puq, z 2 ptq ´z2 puqy and I ∆ s " xz 1 ptq´z 1 puq, z 2 ptq´z 2 puq, z 3 ptq´z 3 puq, . . . , z m`2 ptq´z m`2 puqy have the same integral closure in Ctt, uu and so by remark 1.4.2-5 the ideals θ ´1pI ∆ s q and θ ´1pI ∆ s 2 q have the same integral closure in O X s ˆX s ,p0,0q , which is what we wanted.

In the reducible case the proof works exactly the same way, it is just a lot messier to write down. The only thing you have to keep track off is the following. Suppose pX, 0q has two irreducible components pX 1 , 0qYpX 2 , 0q then pX s , 0q also has two irreducible components and O X,0 -Ctt 1 u ' Ctt 2 u. This implies that the normalization map η s : O X s ,0 ãÑ O X,0 is given by Ctz

1 , . . . , z m`2 u I ãÝÑ Ctt 1 u ' Ctt 2 u z i Þ Ñ pz i pt 1 q, z i pt 2 qq i " 1, 2 ; z j`2 Þ Ñ pz j`2 pt 1 q, z j`2 pt 2 qq j " 1, . . . m In this case O X,0 b C O X,0 -Ctt 1 , u 1 u ' Ctt 1 , u 2 u ' Ctt 2 , u 1 u ' Ctt 2
, u 2 u and the induced morphism θ looks like:

θ : O X s ˆX s ,p0,0q " Ctz 1 , . . . , z m`2 , w 1 , . . . , w m`2 u Ipzq `Ipwq ãÝÑ O X,0 b C O X,0 z i Þ Ñ pz i pt 1 q, z i pt 1 q, z i pt 2 q, z i pt 2 qq i " 1, 2 z j`2 Þ Ñ pz j`2 pt 1 q, z j`2 pt 1 q, z j`2 pt 2 q, z j`2 pt 2 qq j " 1, . . . m w i Þ Ñ pz i pu 1 q, z i pu 2 q, z i pu 1 q, z i pu 2 qq i " 1, 2 w j`2 Þ Ñ z j`2 pu 1 q, z j`2 pu 2 q, z j`2 pu 1 q, z j`2 pu 2 qq j " 1, . . . m,
then you have the map α as in the proof of Proposition 1.4.16 and the rest follows through. Remark 1.4.21 1. Since the Lipschitz saturation map ζ : pX s , 0q Ñ pX, 0q is a generic projection the multiplicity of pX s , 0q is equal to the multiplicity of pX, 0q. 2. Except if the plane branch pX, 0q is non singular, the map pX, 0q Ñ pX, 0q is never obtained as a generic projection since the multiplicity changes. How-ever, among all germs pX 1 , 0q which dominate pX, 0q and are dominated by pX, 0q, and in addition are such that the map pX 1 , 0q Ñ pX, 0q can be represented by a generic linear projection, there is a unique one, up to isomorphism, which dominates all the others: it is the saturation.

Corollary 1.4.22 Let pX, 0q Ă pC 2 , 0q be a reduced plane curve. The Lipschitz saturation map ζ : pX s , 0q Ñ pX, 0q is a biLipschitz homeomorphism.

Proof We already know that a generic projection induces a homeomorphism with its image (Prop. 1.3.3), so by Proposition 1.4.20 the map ζ is a homeorphism and since it is the restriction to X s of the linear projection pz 1 , . . . , z m`2 q Þ Ñ pz 1 , z 2 q it is Lipschitz. The inverse of ζ can be described on each irreducible component X k by

pz 1 pt k q, z 2 pt k qq Þ Ñ pz 1 pt k q, z 2 pt k q, z 3 pt k q, . . . , z m`2 pt k qq,
and since for all j P t1, . . . , mu, z j`2 ptq P O s X,0 Proposition 1.4.16 tells us that it is also Lipschitz.

Őur main result now follows from the following result.

Theorem 1.4.23 (See [P-T69, §4], [B-G-G80, Prop. VI.3.2]) Let O X,0 be the analytic algebra of a germ of reduced plane curve pX, 0q Ă pC 2 , 0q. The Lipschitz saturation O s X,0 determines and is determined by the characteristic exponents of its branches (irreducible components) and the intersection multiplicities m i j " pX i , X j q of each pair of branches. In particular the saturated curve pX s , 0q is an invariant (up to isomorphism) of the equisingularity class of pX, 0q. This implies that every member of the equisingularity class of a germ of reduced plane curve pX, 0q Ă pC 2 , 0q can be obtained by a generic projection of the Lipschitz saturation pX s , 0q of any one of them. The proof of the Proposition involves a lot of calculations and can be found in the references. For this reason we would rather describe how to calculate the saturated curve pX s , 0q. Let us start with the irreducible case: Definition 1.4.24 Let h P Cttu be a power series with coprime exponents. If h " 8 ÿ j"0 a j t j , we define the set of exponents of f as E xp f q " t j P N | a j ‰ 0u. And for a germ of analytically irreducible plane curve pX, 0q Ă pC 2 , 0q we define the set of exponents of O X,0 as

EpO X,0 q " ď hPm E xphq,
where m is the maximal ideal of O X,0 . Note that the semigroup ΓpXq of the plane branch pX, 0q is contained in EpO X,0 q.(See [Tei07, Section 8]).

If pX, 0q Ă pC 2 , 0q is irreducible then: 1. For every j P EpO X,0 q we have that t j P O s X,0 . 2. The analytic algebra O s X,0 is monomial, in particular:

O s X,0 " Ctt j | j P EpO s X,0 qu.

For a numerical semigroup (i.e., a subsemigroup of pN, `q with finite complement) there is the concept of saturated semigroup (see [Ro-G09, Chapter 3, Section 2]) which is defined as follows:

For A Ă N and a P Azt0u define

d A paq " gcdtx P A | x ď au.
Then a non-empty subset A Ă N such that 0 P A and gcdpAq " 1 is a saturated numerical semigroup if and only if a `kd A paq P A for all a P Azt0u and k P N.

The reader can verify that the condition indeed implies that A is a semigroup and that the intersection of two saturated semigroups is again saturated, so that any A Ă N such that 0 P A and gcdpAq " 1 is contained in a smallest saturated semigroup.

Example 1.4.25 Let pX, 0q Ă pC 2 , 0q be the cusp singularity defined by y 2 x3 " 0. Its normalization map is t Þ Ñ pt 2 , t 3 q and so its semigroup is generated by x2, 3y. Since ΓpXq " Nzt1u then EpO X,0 q " ΓpXq is a saturated numerical semigroup.

This definition also tells us how to obtain the smallest saturated semigroup containing any A Ă N with 0 P A and gcdpAq " 1, for example the set of exponents EpO X,0 q. Let e 0 " β 0 " min tx P Au and define

Ă A 0 :" A Y β 0 ¨N.
In the case of EpO X,0 q we have that e 0 " β 0 is the multiplicity of pX, 0q. Let β 1 :" mintx P A | e 0 does not divide xu and e 1 " gcdt β 0 , β 1 u; note that e 1 " d A p β 1 q. Again define

Ă A 1 :" Ă A 0 Y t β 1 `ke 1 | k P Nu.
Continuing this way we obtain two sequences of natural numbers e 0 ą e 1 ą ¨¨¨ą e g " 1 " gcdpAq and β 0 ă β 1 ă ¨¨¨ă β g and an associated sequence of subsets of Ă A 0 Ă Ă A 1 Ă ¨¨¨Ă Ă A g Ă N where β i`1 :" mintx P A | e i does not divide xu, e i :" gcdt β 0 , . . . , β i u " d A p β i q and

Ć A i`1 :" Ă A i Y t β i`1 `ke i`1 | k P Nu.

Note that r

A :" Ă A g is a saturated semigroup which is completely determined by its characteristic sequence t β 0 , . . . , β g u. Moreover if t Þ Ñ `tn , ř iěn a i t i ˘is a Puiseux parametrization of the plane branch pX, 0q Ă pC 2 , 0q, the characteristic sequence of EpO X,0 q is the set of characteristic exponents of pX, 0q and so it determines its equisingularity class.

Proposition 1.4.26 (Pham-Teissier), see [P-T69, §4], [B-G-G80, Thm VI.1.6] For a germ of irreducible plane curve singularity pX, 0q Ă pC 2 , 0q the Lipschitz saturation O s X,0 is given by O s X,0 " Ctt p | p P Č EpO X,0 qu.

In particular E ´Os X,0

¯" Č EpO X,0 q.

Let us give a sketch of the proof: we start from a structured parametrization pt n , yptqq of our branch X as in subsection 1.0.2 and we have to study integral dependence over the ideal I ∆ " pt ´uqN :" xt n ´un , yptq ´ypuqy Ă Ctt, uu.

Here N is the primary ideal x t n ´un t´u , yptq´ypuq t´u yCtt, uu. According to what we saw after Theorem 1.4.5, to verify that gptq ´gpuq is integral over I, which is the same as gptq´gpuq t´u being integral over N , it suffices to verify that its order along any of the branches of a plane curve C T Ă C 2 defined by T 1 t n ´un t´u T2 yptq´ypuq t´u " 0 is larger than that of the ideal I for T " pT 1 : T 2 q in the open set U Ă P 1 . Now we claim that the open set U is T 1 ‰ 0. Indeed, since the order of yptq is ą n all the plane curves C T with T 1 ‰ 0 have a tangent cone consisting of n ´1 lines in general position. It is not difficult then to show (see [Tei73, Chap. II, Lemma 2.6, Proposition 2.7]) that they are equisingular with their tangent cone, and therefore are all equisingular, with simultaneous normalization. So the curve t n ´un t´u " 0 is in U, and its branches are the lines u " ωt, ω P µ n zt1u, which means that a function gptq P Cttu is in the saturation if and only if we have ord t pgptq ´gpωtqq ě ord t pyptq ´ypωtqq for all ω P µ n zt1u.

The result now follows easily from what we saw at the end of subsection 1.0.2 about the orders of the yptq ´ypωtq. It may be interesting to remark here that this construction gives an intrinsic (coordinate free) definition of the Puiseux characteristic as the set of valuations (orders of vanishing) of the ideal N along the irreducible components of the exceptional divisor of the normalized blowing up of N in X ˆX. For more details, see [P-T69, §4] and [B-G-G80, Thm VI.1.6].

Remark 1.4.27 It is shown in [Tei80, 5.2] that the multiplicity, in the sense we saw after Theorem 1.4.7, of the primary ideal N is equal to twice the invariant δ which appears in Propositions 1.2.9 and 1.3.6. It is also shown there that δ is the maximum number of different singular points (then necessarily ordinary double points) which can appear when deforming the parametrization of the plane branch. Both results extends to reducible curves. One can define an analogous ideal N for a non-plane branch but then, in view of Theorem 1.4.8 and Proposition 1.4.11, its multiplicity is twice the δ invariant of a generic plane projection and no longer the classical dim C O X,0 O X,0 in this case.

Example 1.4.28 Let pX, 0q Ă pC 2 , 0q be the plane branch with normalization map: η : pC, 0q ÝÑ pX, 0q t Þ Ñ pt 4 , t 6 `t7 q Then the exponent set EpO X,0 q contains the semigroup ΓpXq " x4, 6, 13yN but by definition it also contains 7. Now β 1 " 6 and e 1 " 2 so Ă E 1 " EpO X,0 q Y t6 `2k | k P Nu.

In the next step β 2 " 7 and e 2 " 1 so g " 2 and we get the saturated semigroup

Ă E 2 " Ă E 1 Y t7 `k | k P Nu.
Note that Č EpO X,0 q " x4, 6, 7, 9yN and so we have the normalization map for the Lipschitz saturation pX s , 0q Ă pC 4 , 0q given by: η s : pC, 0q ÝÑ pX s , 0q t Þ Ñ pt 4 , t 6 , t 7 , t 9 q By making the change of coordinates in pC 4 , 0q, px, y, z, wq Þ Ñ px, y `z, z, wq we can view the Lipschitz saturation map ζ : pX s , 0q Ñ pX, 0q as the projection on the first two coordinates as before.

Remark 1. 4.29 (see [Tei82,Chap. I,Theorem 6.3.1], [B-G-G80, Appendice]) A more concrete way of seeing that all plane branches with the same Puiseux characteristic are generic plane projections of a single space curve is to go back to the notations of subsection 1.0.2 to write down explicitly the saturation of a plane branch pX, 0q with given characteristic pn, β 1 , . . . , β g q: it is isomorphic to the monomial curve with analytic algebra Ctt n , t 2n , . . . , t β1 , t β1`e1 , . . . , t β2 , t β2`e2 , . . . , t β3 , . . . , t βg , t βg `1, . . .u, where n " e 0 " β 0 as above. The semigroup generated by these exponents, which are those of Č EpO X,0 q, is finitely generated by Dickson's Lemma and because the Puiseux exponents are coprime its complement in N is finite. For more details on the saturation of semigroups we refer to [Ro-G09, Chapter 3, Section 2]. As we saw in subsection 1.0.2, up to isomorphism, the image of this monomial curve by a generic linear projection can be parametrized by x " t n , y " ř pP Č EpO X,0 qztnu a p t p . Now we see that the generic projections are precisely those which are such that the coefficient of t n is ‰ 0 and for p " β 1 , . . . , β g we have a p ‰ 0, which means that the projection has characteristic pn, β 1 , . . . , β g q.

Remark that, except if n " 2, the semigroup of integers generated by the exponents of the belonging to the saturation O s X,0 is different from the semigroup Γ we saw in subsection 1.0.2. This has the interesting consequence that the specialization of a plane branch to the monomial curve with the same semigroup, which is Whitney equisingular (see [Gi-T18, Remark 4.1]; the argument there can be generalized to any plane branch), is not in general biLipschitz trivial.

When pX, 0q is not irreducible it is a bit more complicated, nevertheless the Lipschitz saturation O s X,0 can be described in the following way:

Theorem 1.4.30 (see [P-T69, §4] and [B-G-G80, Thm VI.2.2]) Let O X,0 be the analytic algebra of a reduced plane curve pX, 0q " pX 1 , 0q Y . . . Y pX r , 0q with normalization O X,0 " Ctt 1 u ' ¨¨¨' Ctt r u. We may assume that the image of x in O X,0 is `tn 1 1 , . . . , t n r r ˘where n i is the multiplicity of the branch pX i , 0q. Let µ be the least common multiple of tn 1 , . . . , n r u. Then the element h " ph 1 , . . . , h r q P O X,0 is in the Lipschitz saturation O s X,0 if and only if the following two conditions are satisfied: 1. For every j P t1, . . . , ru we have that h j P O s X j ,0 . 2. For every µ-th root of unity and every couple i ‰ j we have the inequality m i, j, phq ě m i, j, :" inf gPO X,0 ! ν τ ´gi pτ µ{n i q ´gj pr τs µ{n j q ¯) ,

where m i, j, phq " ν τ `hi pτ µ{n i q ´hj pr τs µ{n j q ˘and ν τ is the valuation of Ctτu given by the order of the series. The number m i, j, depends only on the characteristic exponents and the intersection multiplicity of the branches X i and X j .

Example 1.4.31 Let pX, 0q " pX 1 , 0q Y pX 2 , 0q be the plane curve with normalization map:

η : pC, 0q \ pC, 0q ÝÑ pX, 0q

t 1 Þ Ñ pt 4 1 , t 6 1 `t7 1 q t 2 Þ Ñ pt 3 2 , t 5 2 q
dense open set U in the space GpN, N ´2q of linear projection, this critical locus is either empty or a reduced curve. The closure of this curve in X is an (absolute) polar curve of X and is denoted by P d´1 pX, πq where d is the dimension of X. It is also denoted by P d´1 pX, Dq, where D " kerπ. These curves play an important role in the local study of singularities, and especially in the study of the Lipschitz geometry of surfaces. See [L-T81], [Tei82, Chap. IV] and [N-P1] for more details.

Of course, if it is not empty, P d´1 pX, πq varies with the projection π P U and a priori it could be that π remains constantly a non generic projection for P d´1 pX, πq. That seems unlikely but still we need a proof for the following: Theorem 1.4.33 (See [Tei82, Chap. V, Lemme 1.2.2]) Given pX, 0q Ă pC N , 0q as above and assuming that P d´1 pX, πq is a reduced curve for π P U Ă GpN, N ´2q, there exists a non empty Zariski open set V Ă U such that for π P V , the projection π : C N Ñ C 2 is a generic projection for the curve P d´1 pX, πq Ă C N .

The proof, which we only sketch, gives an example of the notion of Lipschitz equisaturation , which is found in [P-T69, §6]. Fixing coordinates z 1 , . . . , z N on C N and x, y on C 2 , we can parametrize by C 2pN ´2q a dense open set of the space of linear projections C N Ñ C 2 as follows:

x " z 1 `N ÿ 3 a i z i , y " z 2 `N ÿ 3 b i z i , pa, bq P C 2pN ´2q .

To simplify notations while keeping the ideas, we assume that X is a hypersurface defined by f pz 1 , . . . , z N q " 0. One can also consult [B-H80, Lemme 3.7] which gives the proof for isolated singularities of surfaces in C 3 . For any series hpz 1 , . . . , z N q P Ctz 1 , . . . , z N u let us denote by h a, b the series h a, b pz, a, bq " hpx ´N ÿ 3 a i z i , y ´N ÿ 3 b i z i , z 3 , . . . , z N q.

The equation f a, b " 0 defines a germ of hypersurface Z in C N ˆC2pN´2q and if we consider thefamily of projections π a, b : C N ˆC2pN´2q Ñ C 2 ˆC2pN´2q defined by

x Þ Ñ z 1 `N ÿ 3 a i z i , y Þ Ñ z 2 `N ÿ 3 b i z i , a Þ Ñ a, b Þ Ñ b,
and the closure of its critical locus on the non singular part of Z, we obtain a subspace which, over a Zariski open subset of C 2pN ´2q , contains the family of polar curves associated to the family of projections π a, b . Over a possibly smaller Zariski open subset V of C 2pN ´2q this family of curves is equisingular in the sense that it has a simultaneous parametrization. The number r of irreducible components of Z at points of t0u ˆV Ă C N ˆV is constant and after choosing as origin of C 2pN ´2q a point of V we can parametrize each irreducible component in a neighborhood of t0u ˆt0u by: z 1 " t n , z 2 " υpt , a, bq, z i " ζ i pt , a, bq, with υpt , a, bq, ζ i pt , a, bq P Ctt , a, bu for i " 3, . . . , N. The normalization of O Z,0 being O Z,0 " ś r i"1 Ctt , a, bu. By definition of Z we have for each " 1, . . . , r the identity in Ctt , a, bu Bz 2 does not vanish because it would entail a lack of C 3 transversality (see Definition 1.3.1) of the polar curve with the kernel of the projection which defines it. So we must have on Z the identity z k " Bυ Bb k . By [Tei82, Proposition 6.4.2], after perhaps shrinking V to a smaller Zariski open dense subset V 1 of C 2pN ´2q we have that over V 1 the family Z 1 of plane curves given parametrically by the parametrizations x " t n , y " υpt , a, bq, which consists of the plane projections of our polar curves, is equisaturated. This implies that the derivations B Bb k of Cta, bu extend to derivations D k of O Z 1 ,p0, vq " O Z,p0, vq " ś r i"1 Ctt , a, bu into itself which preserve the relative saturated ring (see [P-T69]). Since of course the functions υpt , a, bq belong to the relative saturation of O Z 1 ,p0, vq , so do the ζ k pt , a, bqq which are their images by D k . But ζ k belonging to this relative saturation means precisely that for v P V 1 , the saturations of the rings O Z 1 pvq and O Zpvq of the fibers over v of Z and Z 1 are equal for v P V 1 , which is the condition for C 5 genericity according to Proposition 1.4.11.

The fact that the plane projection of a generic polar curve by the map which defines it is generic plays an important role in the following three do-

  Remark 1.2.11 We note that one can use Mond-Pellikaan's algorithm in [Mo-P89] to find a presentation matrix of a finite analytic map germ g : pX, 0q Ñ pC d`1 , 0q, where pX, xq is a germ of Cohen-Macaulay analytic space of dimension d. For the computations one can use also the software Singular [D-G-P-S] and the implementation of Mond-Pellikaan's algorithm given by Hernandes, Miranda, and Peñafort-Sanchis in [H-M-P18]. At the web page of Miranda [Mir19] one can find a Singular library to compute presentation matrices based on the results of [H-M-P18].

  0q be a germ of reduced analytic curve, and let Ω Ă GpN 2, Nq be the non-empty Zariski open set of the Grassmannian of pN ´2q-planes of C N which are transversal to C 5 pX, 0q. Then: a) For H P Ω the plane curve pπ H pXq, 0q is reduced and of constant topological (equisingularity) type with Milnor number µ 0 . b) If H R Ω then one of the following statements is verified:

  Remark 1.4.10 See [Gr-P07, Def 1.28, Example 1.46.1 & Lemma 1.89] and [Ada12].

f

  pt n ´N ÿ 3 a i ζ i pt , a, bq, υpt , a, bq´N ÿ 3 a i ζ i pt , a, bq, ζ 3 pt , a, bq, . . . , ζ N pt , a, bqq " 0.Differentiating f a,b " 0 with respect to z i gives the following equations on Z:´ai B f a, b Bz 1 ´bi B f a, b Bz 2 `B f a,b Bz i "0, for i " 3, . . . , N. which by definition are satisfied on the polar curve. Differentiating the first identity with respect to b k and taking into account the second set of identities, we obtain that the equation ˆζk pt , a, bq ´Bυpt , in each Ctt , a, bu. By general transversality results found in [L-T81, Cor. 4.1.6] and [Tei82, Chap. IV, 5.1], B f a, b

  • An element s P S is called integral over R if and only if it satisfies an equation s h `a1 s h´1 `¨¨¨`a h´1 s `ah " 0 with all a i P R. The extension is called integral if every element s P S is integral over R. (Just as in field theory, if the extension R Ă S is finite it is integral. See [De-P00, Lemma 1.5.2]) • The ring R is said to be integrally closed in S if every element in S which is integral over R already belongs to R. • The ring R is called normal if it is reduced and integrally closed in its total quotient ring QpRq.

For more details on what follows in this section, we refer the reader to [Tei07].1 biLipschitz geometry of complex curves: an algebraic approach

biLipschitz geometry of complex curves: an algebraic approach

This precision refers to a distinction between absolute and relative polar varieties, which is not relevant here but should be mentioned to avoid confusions. See [Tei82, Chap. IV, p.

417]

In the previous example we already calculated the Lipschitz saturation O s X 1 ,0 " Ctt 4 1 , t 6 1 , t 7 1 , t 9 1 u and following the algorithm we get the Lipschitz saturation O s X 2 ,0 " Ctt 3 2 , t 5 2 , t 7 2 u. Since the branches are tangent, their intersection multiplicity is greater than the product of their multiplicities and is equal to the order of the series in t 1 obtained by substituting the parametrization of pX 1 , 0q in the equation y 3 ´x5 " 0 defining pX 2 , 0q. In this case it is equal to 18.

By definition µ " lcmt3, 4u " 12 and it is not hard to prove that for any 12´th root of unity m 1,2, " ν τ `y1 pτ 3 q ´y2 pr τs 4 q " ν τ `τ18 `τ21 ´ 8 τ 20 ˘" 18.

So from the Theorem 1.4.30 we have that h " ph

and m 1,2, phq ě 18. For example if h " pt 4 1 , t 5 2 q then m 1,2, phq " ν τ `pτ 3 q 4 ´pr τs 4 q 5 " ν τ pτ 12 ´ 8 τ 20 q " 12 ñ h R O s X,0 .

On the other hand if h " pt 6 1 `t7 1 , t 5 2 q then m 1,2, phq " ν τ `pτ 3 q 6 `pτ 3 q 7 ´pr τs 4 q 5 " ν τ pτ 18 `τ21 ´ 8 τ 20 q " 18 ñ h P O s X,0 .

We will end this section with the following consequence of the Theorem:

Corollary 1.4.32 (see [B-G-G80, VI.3.7]) Let pX, 0q " pX 1 , 0q Y . . . Y pX r , 0q be a reduced plane curve with normalization O X,0 " Ctt 1 u ' ¨¨¨' Ctt r u. If Π j : O X,0 Ñ Ctt j u denotes the canonical projection to the j´th factor then Π j pO s X,0 q " O s X j ,0 .

Application to local polar curves

Let pX, 0q Ă pC N , 0q be a reduced equidimensional germ of complex analytic space. Consider linear projections π : C N Ñ C 2 and the critical locus of π restricted to the smooth part X 0 of X. It is proved in [L-T81], where the theory of (absolute)