Dmitri Korotkin
email: korotkin@mathstat.concordia.ca

Adrien Sauvaget
email: adrien.sauvaget@imj-prg.fr

Peter Zograf
email: zograf@pdmi.ras.ru

TAU FUNCTIONS, PRYM-TYURIN CLASSES AND LOCI OF DEGENERATE DIFFERENTIALS

Keywords: Mathematics Subject Classification. 14H15, 14F10, 14H70, 30F30, 14C22 Moduli space of curves, n-differentials, cyclic covers, Bergman tau function, integrable systems

We study the rational Picard group of the projectivized moduli space P M (n) g of holomorphic n-differentials on complex genus g stable curves. We define n -1 natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differentials with a double zero. We give two different proofs of this result, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the universal curve.

Moduli space of n-differentials. Let g and n be positive integers with g ≥ 2. Let M g (respectively M g ) be the moduli space of smooth (respectively stable nodal) complex curves. Denote by D 0 ⊂ M g the closure of the locus of stable curves with one nonseparating node. Further, denote by D i ⊂ M g , 1 ≤ i ≤ [g/2], the closure of the locus of curves with a separating node and two irreducible components of genera i and g -i. Finally, denote by π : C g → M g or π : C g → M g the universal curve and by ω = ω Cg/Mg the relative dualizing sheaf.

Let Ω (n) g = R 0 π * ω ⊗n → M g be the direct image of the nth tensor power of ω. Using the Riemann-Roch formula and Serre's duality, one can easily check that h 1 (C, ω ⊗n C ) = 0 if n ≥ 2 and h 1 (C, ω C ) = 1 for any stable curve C. Thus Ω (n) g is a vector bundle for any n ≥ 1.

For n = 1 we write Ω g instead of Ω [START_REF] Babelon | Introduction to classical integrable systems[END_REF] g and call it the Hodge bundle. The Riemann-Roch formula implies:

rk Ω (n) g = g if n = 1; (2n -1)(g -1) otherwise.

We define the following elements in the rational Picard group of M g :

• the Chern class λ n ∈ Pic(M g ) of the determinant line bundle of Ω For k = 1 we write λ instead of λ 1 and call this class the Hodge class.

Definition 1.1. The total space of the vector bundle Ω

(n) g is denoted by M (n) g and is called the space of n-differentials.

The points of M

(n) g correspond to equivalence classes of pairs (C, w), where C is a stable genus g algebraic curve, and w is an n-differential on C. We recall (see [START_REF] Bainbridge | Strata of k-differentials[END_REF] for example) that an n-differential w on C is a meromorphic n-differential on each irreducible component of the normalization of C such that • w can only have poles at the preimages of the nodes; • these poles are of order at most n;

• at every node the n-residues of w at the poles satisfy res p1 (w) = (-1) n res p2 (w) where p 1 and p 2 are the two preimages of the node.

We denote by ν : M (n) g → M g the forgetful map and we will use the same notation for its restriction ν : M (n) g → M g to the locus of smooth curves. We also denote by ν : P M (n) g → M g the projectivized space of n-differentials.

In this paper we study the Picard group of P M (n) g . We will work over rational numbers, so that Pic will always denote the rational Picard group.

By abuse of notation we will denote by λ, λ n and δ i both the elements of Pic(M g )

and their pull-backs in Pic(P M

(n) g ). In addition, we introduce the first Chern class ψ ∈ Pic(P M (n) g ) of the tautological line bundle L → P M (n) g . The following lemma is standard (cf., e.g, [START_REF] Korotkin | Tau function and moduli of differentials[END_REF], Lemma 1).

Lemma 1.2. The classes λ, ψ, δ 0 , . . . , δ [g/2] form a basis of Pic(P M (n) g ).

The goal of this paper is to define the Prym-Tyurin classes in Pic(P M (n) g ) and express them in the above basis.

Stratification of P M

(n) g . The space of n-differentials is naturally stratified according to the multiplicities of the zeroes of the differential.

Let k = (k 1 , . . . , k m ) be a partition of n(2g -2). We denote by M (n)

g [k] ⊂ M (n) g
the locus of pairs (C, w) such that the n-differential w has m pairwise distinct zeros of orders exactly k i . This locus is C * -invariant, thus we can also define its projectivization P M

(n)

g [k] ⊂ P M (n)
g . The space P M

(n) g

is the disjoint union of the strata P M (n) g [k] for all partitions k of n(2g -2). The following properties of the strata were proved in [START_REF] Lanneau | Connected components of the strata of the moduli spaces of quadratic differentials[END_REF] and [START_REF] Schmitt | Dimension theory of the moduli space of twisted k-differentials[END_REF].

• Each stratum P M (n) g [k] is smooth.
• If at least one k i is not divisible by n then either P M We denote by P M

(n) g [k] the closure of P M (n) g [k] in P M (n) g . In particular we have P M (n) g [1] = P M (n)
g , where 1 stands for the partition (1, 1, . . . , 1). Definition 1.3. Let g, n ≥ 2. The divisor of degenerate n-differentials is defined as

D deg = P M (n) g [2, 1, . . . , 1], if (g, n) = (2, 2), P M (n) g [2, 1, 1] + 2 • P M (n) g [2, 2], if g = n =
, n) = (2, 2) it is just the closure of P M (n) g [2, 1, . . . , 1] in P M (n)
g . In the case g = n = 2, however, D deg has a special component consisting of squares of holomorphic differentials. This is because in genus 2 each quadratic differential is invariant with respect to the hyperelliptic involution. The four simple zeroes of w are pairwise equivalent under the hyperelliptic involution, and when two nonequivalent zeroes coalesce, the other two ones also coalesce, giving a differential with two double zeroes. Since every such differential has two square roots that differ by a sign, the divisor P M (n) g [START_REF] Bainbridge | Strata of k-differentials[END_REF][START_REF] Bainbridge | Strata of k-differentials[END_REF] comes with a factor of 2. (Note that when two equivalent zeroes coalesce, the differential in the limit has one double zero at a Weierstrass point and two simple zeroes.) 1.3. First definition of Prym-Tyurin classes. Let (C, w) be a point in the projectivized moduli space P M (n) g [START_REF] Babelon | Introduction to classical integrable systems[END_REF]. One can define a canonical cyclic ramified cover f : C → C of degree n, where

C = {(x, v)|x ∈ C, v ∈ T *
x C, v n = w}. This cover is completely ramified over the zeros of w. The curve C is smooth of genus g = n 2 (g -1) + 1. It comes with a canonical holomorphic differential v given by v(x, v) = v. This differential v on C satisfies v n = f * w.

The action of Z/nZ on the cover is given by ρ

k : (x, v) → (x, ρ k v), where ρ = e 2π √ - 1 n
. We denote by σ : C → C the automorphism of C corresponding to k = 1. Now consider the natural map

ν : P M (n) g [1] → M g , (C, w) → C
( C remains the same when we multiply w by a non-zero constant). We consider the pull-back of the Hodge bundle Ω g by the map ν. The automorphism σ induces an endomorphism σ * of the vector bundle ν * Ω g given by: ((C, w), u) → ((C, w), σ * u), where u is an element of H 0 ( C, ω C ). The endomorphism σ * satisfies (σ * ) n = Id. Hence we have a decomposition

(1.1) ν * Ω g = n-1 k=0 Λ (k) ,
where Λ (k) is the eigenbundle of ν * Ω g corresponding to the eigenvalue

ρ k = e 2π √ -1k n .
Remark 1.5. The space Λ (k) is a vector bundle because the dimension of the fiber of Λ (k) is upper semi-continuous for all k and ν * Ω g is a vector bundle thus the rank of each Λ (k) is constant. We will see further that rank(Λ (k) ) = (2n -2k + 1)(g -1).

Definition 1.6. The vector bundles Λ (k) are called the Prym-Tyurin vector bundles. The Prym-Tyurin class λ

(k) P T is the first Chern class c 1 (Λ (k) ) ∈ Pic(P M (n)
g [START_REF] Babelon | Introduction to classical integrable systems[END_REF]). For n = 2 the study of vector bundles of this type was initiated by Prym [START_REF] Prym | Einer allgemeinen Thetaformel[END_REF] and for n > 2 by A. N. Tyurin [START_REF] Tyurin | Vector bundles. Collected works[END_REF]. we can construct the object (f : C → C, σ) and the eigenspaces of σ * , however the genus of C and the dimensions of the eigenspaces of σ * will depend on the orders of zeros of w. We will even prove in Section 5 that the map ν : M (n) g → M ĝ , used to define the Prym-Tyurin vector bundles, admits no natural extension to P M (n) g . Nonetheless, in the next section we extend the definition of the Prym-Tyurin class to P M (n) g by a construction involving the space of admissible covers. This will allow to compactify the P M (n) g in such a way that the cover constructed above is of constant arithmetic genus.

1.4. Admissible covers and Prym-Tyurin bundles. We denote by Hur n g the moduli space whose geometric points are isomorphism classes of pairs (f : C → C, σ) where:

• C and C are smooth curves; • f is a cyclic ramified cover of degree n which is totally ramified over N := deg(ω ⊗n ) = 2n(g -1) distinct points of C; • σ is an automorphism of C that commutes with f . We denote by Hur n g the compactification of this space by admissible covers (see [START_REF] Harris | On the Kodaira dimension of the moduli space of curves[END_REF]). The space of admissible covers has two forgetful maps (source and target of the cover):

Hur n g target z z source " " M g,N /S N M g .
We consider the pull-back source * Ω g of the Hodge bundle under the source map. This vector bundle is endowed with the automorphism

σ * : ( C → C, σ), u → ( C → C, σ), σ * u .
Thus, as in (1.1), we have the decomposition

(1.2) source * Ω g = n-1 k=0 Λ (k) ,
where Λ (k) is the eigenbundle corresponding to the eigenvalue ρ k = e 2π √ -1k n . In the previous section we have constructed an embedding i :

P M (n) g [1] → Hur n g
and, by construction, the pull-back i * Λ (k) is, indeed, isomorphic to the Prym-Tyurin bundle Λ (k) over P M (n) g [START_REF] Babelon | Introduction to classical integrable systems[END_REF] as defined in the previous section. We will see that the Prym-Tyurin vector bundles, and therefore the Prym-Tyurin classes, have a natural extension to the compactification of P M (n) g [START_REF] Babelon | Introduction to classical integrable systems[END_REF] by admissible covers. We would like, however, to extend the Prym-Tyurin classes to a different compactification of P M (n) g [START_REF] Babelon | Introduction to classical integrable systems[END_REF], namely, to P M (n) g . To do that we will construct a bigger space with a projection to both Hur

M (n) g [1] → M (n) g × Mg Hur n g
be the product of the two natural embeddings. The moduli space of admissible n differentials X(g, n), is the Zariski closure of the image of I in M

(n) g × Hur n g . Remark 1.9. The space of admissible n-differentials is a compactification of the stratum M (n) g [START_REF] Babelon | Introduction to classical integrable systems[END_REF]. In [START_REF] Bainbridge | Strata of k-differentials[END_REF], the authors introduced and described another compactification, the incidence variety compactification. The incidence variety is obtained by replacing the space Hur n g in Definition 1.8 by the space M g,N /S N . We denote the incidence variety by X inc g,n . There is a birational and finite map from X(g, n) to X inc g,n , however this map is not an isomorphism (see Example 4.3 of [START_REF] Bainbridge | Strata of k-differentials[END_REF]). We will make use of X inc g,n once in this text (proof of Lemma 2.4).

The space X(g, n) has two natural morphisms: adm : X(g, n) → Hur n g and diff : X(g, n) → P M (n) g . We summarize the notation on the following diagram.

Λ (k) source * / / Ω g X(g, n) adm / / diff Hur n g source / / M g P M (n) g
By construction the forgetful map diff : X(g, n) → P M (n) g is birational and its restriction to P M (n) g [START_REF] Babelon | Introduction to classical integrable systems[END_REF] is an isomorphism onto its image. In general the space X(g, n) is not normal; thus the push-forward of classes in the Picard group under diff is ill-defined. However we will prove the following proposition in Section 2.

Proposition 1.10. There exist two smooth open-dense substacks j : V → X(g, n)

and j : U → P M (n) g fitting into the commutative diagram V j / / X(g, n) diff U j / / P M (n) g such that
• the map V → U is an isomorphism on the underlying coarse spaces;

• the complement of U is of codimension at least 2 in P M (n) g .
The existence of U and V as above allows one to define the induced push-forward diff * in the Picard groups. Indeed, the first property ensures that Pic(U ) and Pic(V ) are isomorphic. Similarly, the second propery ensures that j induces an isomorphism between Pic(U ) and Pic(P M (n) g ). Thus, given an element in Pic(X(g, n)), one defines its push-forward to Pic(P M (n) g ) by first taking its pull-back by j, then the push-forward from V to U , and finally the push-forward by j .

We will see that the definition of diff * is independent of the choice of U and V satisfying the above hypothesis (see Lemma 2.2). Theorem 1.12. In the rational Picard group of P M (n) g we have

(1.3) δ deg = 12n(n + 1)λ -2(g -1)(2n + 1)ψ -n(n + 1) [g/2] i=0 δ i ; (1.4) λ (n-k) P T = (6k 2 + 6k + 1)λ - g -1 n k(2k + 1)ψ - 1 2 k(k + 1) [g/2] i=0 δ i + c k δ deg ,
where

(1.5) c k = 2k-n 2n , if (n -1)/2 < k < n, 0, otherwise. . 1.7.
Strategy of the proof. Formula (1.3) of Theorem 1.12 is proved in two distinct ways.

• In Section 3 we introduce the Bergman tau function on the moduli space M

(n)

g . We study its transformation property and its asymptotic behavior at the boundary divisors D deg and D i , 0 ≤ i ≤ [g/2]. We explicitly compute the vanishing order of the Bergmann tau function along these divisors. We use these results to express the divisor δ deg in the (λ, δ i , ψ) basis of the Picard group. This first proof is a further development of the ideas introduced in [START_REF] Korotkin | Tau function and moduli of differentials[END_REF] and [START_REF] Korotkin | Tau function and the Prym class[END_REF].

• In Section 4 we give an alternative proof of Formula (1.3) based on algebrogeometric computations. We consider the moduli space of n-differentials on genus g curves with one marked point. This space carries a vector bundle of 2-jets of an n-differential at the marked point. The Euler class of this vector bundle has a natural expression involving the locus A 2 of ndifferentials with a double zero at the marked point. The locus A 2 pushes forward to D deg under the forgetful map that forgets the marked point. This allows one to compute the cohomology class δ deg that is Poincaré dual to the divisor class of D deg . To prove Formulas (1.4) and (1.5) we combine (1.3) with the following two facts.

• First, the well-known Mumford formula [START_REF] Mumford | Stability of projective varieties[END_REF] expressing the first Chern class of the vector bundle of k-differentials on M g via the Hodge class:

(1.6)

λ k = (6k 2 -6k + 1)λ - k(k -1) 2 [g/2] i=0 δ i .
• Second, the fact that the morphism

(1.7) Φ k : Λ (k) ⊗ T ⊗n-k → H 0 (C, ω n-k+1 C ) (q, v n-k ) → qv n-k
is actually an isomorphism of vector bundles outside D deg .

The second fact allows one to compute the rank of the Prym-Tyurin vector bundle Λ (k) :

(1.8) rk Λ (k) = rk Ω (n-k+1) = (2n -2k + 1)(g -1) , k = 1, . . . , n -1 .
It also implies that (1.9) λ (k)

P T = λ n-k+1 - g -1 n (n -k)(2n -2k + 1)ψ + const • δ deg .
In Section 5, we study the asymptotic of the determinant of Φ k along the divisor D deg to obtain Expressions (1.4) and (1.5).

Plan of the paper. In Section 2 we prove Proposition 1.10 and thus complete the definition of the Prym-Tyurin classes. In Sections 3 and 4 we prove Formula (1.3) of Theorem 1.12 using the two different approaches described above. In Section 5 we discuss the relationship between Prym-Tyurin vector bundles and vector bundles of n-differentials and derive a relationship between corresponding determinant line bundles. This allows us to express the Prym-Tyurin classes in the (λ, δ i , ψ) basis of the Picard group and complete the proof of Theorem 1.12.

The space of admissible n-differentials

In this Section we justify the definition of the Prym-Tyurin classes by proving the following extended version of Proposition 1.10.

Proposition 2.1. There exist two smooth open-dense substacks

j : V → X(g, n) and j : U → P M (n) g fitting into the commutative diagram V j / / X(g, n) diff U j / / P M (n) g such that • V and U contain the image of P M (n)
g [START_REF] Babelon | Introduction to classical integrable systems[END_REF] under the embeddings into X(g, n) and P M

(n) g ; • the complement of U is of codimension at least 2 in P M (n) g ;
• the map V → U is an isomorphism on the underlying coarse spaces;

• there exists a line bundle T → V such that T is a sub-vector bundle of Λ (1) , T ⊗n L and the restriction of T to P M

(n)

g [START_REF] Babelon | Introduction to classical integrable systems[END_REF] coincides with the sub-vector bundle of ν * Ω g spanned by v.

Before proving this proposition, let us prove the following lemma.

Lemma 2.2. Let (U → P M (n) g , V → X(g, n)) and (U → P M (n) g , V → X(g, n))
be two sub-stacks satisfying the list properties of Proposition 1.10. Let diff * and diff * be the morphisms Pic(X(g, n)

) → Pic(P M (n) g ) constructed as in Section 1.5. Then we have diff * = diff * . Proof. Let us denote by U = U × P M (n) g U and V = V × X(g,n) V . Then the pair (U → P M (n) g , V → X(g, n)
) satisfies the list of properties of Proposition 1.10. Moreover the complement of U in U and U is of co-dimension at least two, thus we have Pic(U ) Pic(U ) Pic(U ) (the same stands for V ). Moreover we have V = V × U U thus the left square on the following diagram is cartesian

V i / / V j / / X(g, n) diff U i / / U j / / P M (n) g and we have i * • diff |V * • i * = diff
(v) = nx 1 + • • • + nx N . Introduce a local parameter ζ i in a neighborhood of x i ∈ C and a local parameter ξ i in a neighborhood of xi ∈ C such that (2.2) w = ζ i (dζ i ) n , ξ i (x) n+1 = x xi v.
Both parameters are defined up to an (n + 1)st root of unity and we make one choice in such a way that

ζ i = n + 1 n n/(n+1) ξ n i .
The local parameters ξ i on C and ζ i on C given by (2.2) are called distinguished.

Since f * v = ρ v, the local parameter ξ i (x) transforms under the action of f as ξ i (f (x)) = ρξ i (x).

Extension of the Prym-Tyurin bundles to codimension 1 loci. By construction, P M (n)

g [START_REF] Babelon | Introduction to classical integrable systems[END_REF] is an open dense substack of X(g, n). Thus diff :

X(g, n) → P M (n)
g is birational and its restriction to P M (n) g [START_REF] Babelon | Introduction to classical integrable systems[END_REF] is an isomorphism onto its image. By abuse of notation we denote by

D i , 0 ≤ i ≤ [g/2], the preimage in P M (n) g of the boundary divisor D i ⊂ M g . Then the complement of P M (n) g [1] in P M (n) g is the union of the divisors D i and D deg .
For each of these divisors we define a dense open locus D ⊂ D as follows.

• D 0 is the locus of (C, w) such that the curve C has exactly one nonseparating node and the differential w has poles of order n at the node and N simple zeros; • D i for i ≥ 1 is the locus of (C, w) such that the curve C has exactly one separating node and the differential w has poles of order n at the node and N simple zeros.

• D deg if (g, n) = (2, 2
) is the locus of (C, w) such that the curve C is a smooth curve and the differential w has one zero of order exactly 2 and its other zeros are simple;

• D deg if (g, n) = (2, 2) is the disjoint union of the locus D deg (2, 1, 1) described
above and of the locus D deg (2, 2) of pairs (C, w) where C is smooth and w is a square of a holomorphic differential with simple zeros.

We define U as the union of P M

(n) g [1] ∪ D deg ∪ i D i ⊂ P M (n)
g . We define V as diff -1 (U ) ⊂ X(g, n). We will prove that U and V satisfy the properties of Proposition 2.1.

Property 1 is satisfied by construction. and its complement is of codimension at least 2.

Proof. The complement of U is a union of closed substacks: the strata of curves with at least two nodes and the strata P M ). This quotient can be realized as a Deligne-Mumford (DM) stack or as a scheme. In the later case, it is isomorphic to Spec(C[u]). We have a map from the DM stack C/(Z/2Z) → C, defined by u → x 2 . Then the divisor {x = 0} is mapped to the divisor {u = 0} with degree 1/2.

Proof. The underlying scheme of U is smooth and the map diff : V → U is birational. Thus the map diff of schemes is an isomorphism if and only if it is finite. We consider the incidence variety X inc g,n compactification defined in Remark 1.9. We have two birational map: diff X inc g,n → X(g, n) and :

X inc g,n → M (n) g
such that diff = diff • . The map diff is obtained by forgetting the admissible cover (but not the markings) and is obtained by forgetting the markings. As we have already stated in Remark 1.9, the map diff is finite because Hur g,n → M g,N /S N is finite. Therefore we need to check that the map restricted to -1 (U ) is finite.

The restriction : -1 (U ) → U is a bijection. Indeed, if (C, w) be a n-differential in U \ D deg , then the preimage of (C, w) under is the n-differential w with the marked simple zeros. Now if (C, w) is a n-differential in Ddeg then the preimage of (C, w) is the point (C , w , x i ) where C is the curve with two components: one component isomorphic to C and one rational component attached to C at the double zero; the differential w is then given by w on the main component and vanishes identically on the rational component; finally the marked points are the simple zeros on the main component and two marked points on the rational component.

Therefore : -1 (U ) → U is finite and diff : V → U is birational and finite thus an isomorphism of the underlying schemes. Moreover the restriction of the map to U \ D deg is obviously an isomorphism of stacks. The degree of diff along D deg will be computed in the next paragraphs.

Lemma 2.6 (Property 4

). There exists a line bundle T → V such that T ⊗n L, T is a sub-vector bundle of Λ (1) , and the restriction of T to P M (n) g [START_REF] Babelon | Introduction to classical integrable systems[END_REF] coincides with the sub-vector bundle of ν * Ω g spanned by v.

The proof requires a detailed analysis of the inverse morphism diff -1 : U → V and is contained in the next two subsections.

2.2.1. Nodal curves. Let 0 ≤ i ≤ [g/2] and let (C, w) be a point in D i . The n-fold cover associated to (C, w) is given by C = {(x, v) ∈ T * C /v n = w} and the canonical differential v is still defined by v(x, v) = v.
We can describe the topology of C and the singularities (zeros and poles) of v:

• If i = 0, then the curve C is an irreducible curve with n self-intersections.

The differential v has zeros of order n at the marked points and poles of order 1 at the nodes. • If i ≥ 1 then C has two irreducible components intersecting at n distinct nodes. The differential v has also zeros of order n at the marked points and poles of 1 at the nodes. The canonical differential v is well-defined on U \ D deg , thus the line bundle T can be extended to diff -1 (U \ D deg ).

Degenerate differentials.

Here we describe the local structure of the stacks X(g, n) and M (n) g close to D deg . This allows us to make explicit the isomorphism of Lemma 2.4 and to describe the fiber of the canonical line bundle along D deg . Let (C 0 , w 0 ) be a point in D deg and let W be a neighborhood of (C 0 , w 0 ) in D deg . We will give a local parametrization of X(g, n) and M (n) g around the point (C 0 , w 0 ).

• Parameters of M (n) g . A neighborhood of (C 0 , w 0 ) is given by W × ∆ where ∆ is a disk in C centered at zero. A point (u, a) in W × ∆ parametrizes an n-differentials (C, w) such that w = (ζ 2 + a)dζ n .
where the parameter ζ of the curve C is uniquely determined by the choice of a. The parameter a is a tranverse local parameter of

D deg in M (n) g . • Parameters of X(g, n). A neighborhood of the point (C 0 , w 0 ) in X(g, n) is parametrized by W × ∆ /(Z/
w(x) = (ζ(x) -ζ 1 )(ζ(x) -ζ 2 )(dζ(x)) n , x ∈ C. The parameter (ζ 1 -ζ 2 )/{±1} is a local transverse parameter to D deg in X(g, n) (See Lemma 3.6 for a proof).
With these two local parametrizations, the map diff :

X(g, n) → M (n) g
is given by

W × ∆ /(Z/2Z) → W × ∆ (u, ζ 1 -ζ 2 ) → (u, (ζ 1 -ζ 2 ) 2 ).
This map is indeed an isomorphism of the underlying schemes. However it is of degree 1/2 along D deg ⊂ X(g, n) once we consider the stack structures of X(g, n)

and M

(n) g . This finishes the proof of Lemma 2.4.

Finally we describe the extension of the canonical line bundle T to D deg . Let (C, w) be a family of differentials with simple zeros which tends to (C 0 , w 0 ) ∈ D deg ⊂ X(g, n). Once again we label the two coalescing zeros x 1 and x 2 and we use the local parameter of the curve

ζ with ζ(x i ) = ζ i and w(x) = (ζ(x)-ζ 1 )(ζ(x)- ζ 2 )(dζ(x)) n
As (C, w) tends to a pair (C 0 , w 0 ) the zeros x 1 and x 2 tend to the double zero x 0 of w 0 . The limit curve C 0 is a nodal curve with two components: a Riemann sphere C 1 which gets naturally equipped with the meromorphic n-differential w 1 (ζ) given by the formula

(2.3) w 1 (ζ) = (ζ -ζ 1 )(ζ -ζ 2 )(dζ) n , ζ ∈ C 1 ,
which is holomorphic outside of ζ = ∞ and has two simple zeros at ζ 1 and ζ 2 . The Riemann surface C 2 is equipped with the holomorphic n-differential w 0 . The nodal point on C 0 is formed by identifying the point ζ = ∞ on C 1 with the point x 0 on C 2 . The limit n-differential (C 0 , w 0 ) determines a canonical n-sheeted cover C 0 → C 0 . The curve C 0 consists of two components C 1 and C 2 . The canonical cover C 1 of C 1 is given by equation

(2.4) v n 1 = w 1 (ζ) which, if we write v 1 = ydζ, is the curve (2.5) y n = (ζ -ζ 1 )(ζ -ζ 2 ) of genus g 1 = [(n -1)/2]. The canonical cover C 2 of C 2 is defined by the equation (2.6) v n 2 = w 2 ; its genus equals g 2 = g -[n/2].
Therefore, for odd n we have g = g 1 + g 2 while for even n we have g = g 1 + g 2 + 1.

The difference between the case of even n and the case of odd n is due to the fact that for odd n the covers C 1 and C 2 intersect at only one nodal point while for even n the nodal point on C 0 has two pre-images on C 0 i.e. for even n C 1 and C 2 intersect at two nodal points, x

(1) 0 and x (2) 0 (see Figure 1 below). With this description of the limit cover, we define the limit canonical differential v 0 as follows: it is given by v 0 = v 2 on the component C 2 and vanishes identicaly on the rational component C 1 . It satisfies v n 0 = f * w 0 . Thus the canonical line bundle T can be extended to the open set V . This completes the proof of Lemma 2.6. Tau functions play an important role in the theory of integrable systems providing canonical generators for commuting flows on the phase space [START_REF] Babelon | Introduction to classical integrable systems[END_REF]. In some cases tau functions carry interesting algebro-geometric information, like the isomonodromic tau function of the Riemann-Hilbert problem that plays a noticeable role in the theory of Frobenius manifolds [START_REF] Dubrovin | Geometry of 2D topological field theories[END_REF].

By means of the Bergman tau function on the Hurwitz space [START_REF] Kokotov | Isomonodromic tau function of Hurwitz Frobenius manifolds and its applications[END_REF], the Hodge class on the space of admissible covers of the complex projective line was explicitly expressed as a linear combination of the boundary divisors [START_REF] Kokotov | Isomonodromic tau function on the space of admissible covers[END_REF]. This result was rederived in [START_REF] Van Der Geer | The Hodge bundle on Hurwitz spaces[END_REF] by pure algebro-geometric methods (namely, using the Grothendieck-Riemann-Roch theorem) and was applied in [START_REF] Van Der Geer | The class of a Hurwitz divisor on the moduli of curves of even genus[END_REF] to answer a question of Harris-Mumford [START_REF] Harris | On the Kodaira dimension of the moduli space of curves[END_REF] about the classes of Hurwitz divisors in the moduli space M g of stable complex algebraic curves of even genus g.

A version of the Bergman tau function for the moduli space of holomorphic abelian differentials on algebraic curves [START_REF] Kokotov | Tau-functions on spaces of Abelian differentials and higher genus generalization of Ray-Singer formula[END_REF] was used in [START_REF] Korotkin | Tau function and moduli of differentials[END_REF] to obtain new relations in the rational Picard group. These results appeared to be relevant to the Kontsevich-Zorich theory of Teichmüller flow, cf. [START_REF] Korotkin | Tau function and moduli of differentials[END_REF], [START_REF] Eskin | Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow[END_REF]. In [START_REF] Korotkin | Tau function and the Prym class[END_REF], Bergman tau functions helped to express the Hodge and Prym classes on the moduli space of holomorphic quadratic differentials in terms of other divisor classes. In this section we continue developing these ideas further for the moduli space of holomorphic n-differentials.

3.1.

Bergman tau function on strata of n-differentials. We begin with defining the Bergman tau function for each stratum

M (n) g [k] where k = (k 1 , . . . , k m ) is a partition of N = 2n(g -1)
. We introduce the following notation (see [START_REF] Fay | Kernel functions, analytic torsion, and moduli spaces[END_REF] for precise definitions):

• v 1 , . . . , v g -the normalized basis of holomorphic abelian differentials with respect to a given Torelli marking (or cut system) on C; • Ω -the corresponding period matrix; • Θ(z, Ω) -the theta function associated with Ω;

• W (x) -the Wronskian determinant of differentials v 1 , . . . , v g ; • C -the fundamental polygon corresponding to the chosen cut system on C; • E(x, y) -the prime form on C × C; • A x -the Abel map corresponding to the initial point x; • K x -the vector of Riemann constants corresponding to the initial point x.

The distinguished local parameters on C in a neighborhood of the points x i (zeroes of the n-differential w) are given by (3.1)

ζ i (x) = x xi v n/(ki+n)
where k i is the order of zero of w at the point x i (in terms of these parameters

w ∼ ζ ki i (dζ i ) n near x i ∈ C, and v = w 1/n ∼ ζ ki/n i dζ i near xi = f -1 (x i ) ∈ C).
The local parameters (3.1) are defined up to an nth root of unity, and different sheets of C correspond to different choices of this root. Then for the prime form E(x, y) on C × C we have

E(x, y) = E(ζ(x), ζ(y)) √ dζ(x) √ dζ(y) ,
and we put

E(ζ, x k ) = lim y→x k E(ζ(x), ζ(y)) dζ k dζ (y), E(x k , x l ) = lim x→x k y→x l E(ζ(x), ζ(y)) dζ k dζ (x) dζ l dζ (y) .
We define two vectors Z, Z ∈ 1 n Z g by the condition

(3.2) 1 n A x ((w)) + 2K x = ΩZ + Z .
Definition 3.1. The Bergman tau function on the space

M (n) g [k] is given by τ (C, w) = (3.3) c(x) 2/3 e -π 6 ΩZ,Z -2π √ -1 3 Z,K x w(x) m i=1 E ki (x, x i ) (g-1)/3n i<j E(x i , x j ) k i k j 6n 2 , where c(x) = 1 W (x) g i=1 v i (x) ∂ ∂z i g θ(z; Ω) z=K x .
From Eq. (3.3) it is clear that the tau function is nowhere vanishing and nonsingular on M

(3.4) τ (C, w, {ã i , bi }) τ (C, w, {a i , b i }) = det(CΩ + D)
where is a root of unity of degree 48d with d = l.c.m.(k 1 + n, . . . , k m + n).

The proof can be obtained by using standard transformation properties of all factors in (3.3) under the change of Torelli cut system on C (cf. [START_REF] Fay | Kernel functions, analytic torsion, and moduli spaces[END_REF]). The root of unity appears due to an ambiguity in the definition of the distinguished local parameters (3.1), which translates into an ambiguity in the definition of E(x, x i ) and E(x i , x j ). The appearance of the term det(CΩ + D) can also be seen from variational formulas for τ (C, w) discussed below, similarly to [START_REF] Kokotov | Tau-functions on spaces of Abelian differentials and higher genus generalization of Ray-Singer formula[END_REF][START_REF] Korotkin | Tau function and moduli of differentials[END_REF][START_REF] Korotkin | Tau function and the Prym class[END_REF].

Proposition 3.3. The tau function has the following quasi-homogeneity property:

(3.5) τ (C, δw) = δ κ τ (C, w) with (3.6) κ = 1 12n 2 m i=1 k i (k i + 2n) k i + n .
This proposition follows from the explicit formula (3.3), but can also be derived by applying the Riemann bilinear identity to variational formulas for τ as it was done in [START_REF] Kokotov | Isomonodromic tau function on the space of admissible covers[END_REF] in the context of Hurwitz spaces.

Combining Propositions 3.2 and 3.3, we arrive at the following Theorem 3.4. On the stratum P M

(n) g = P M (n) g [1] ⊂ P M (n)
g of n-differentials with simple zeroes, the power τ 48n(n+1) of the tau function τ = τ (C, w) is a nowhere vanishing holomorphic section of the line bundle λ 48n(n+1) ⊗ L -8(g-1)(2n+1) -→ P M (n) g .

In order to find the divisor of the section τ 48n(n+1) on P M (n) g , we will compute the asymptotics of τ at the boundary divisors D deg and D j , j = 0, 1, . . . , [g/2]. For that we need to study the tau function more carefully.

3.2. Homological coordinates and variational formulas for the tau function. The tau function τ (C, w) satisfies a system of linear differential equations on the space M (n) g similar to the tau functions on Hurwitz spaces, or spaces of abelian or quadratic differentials [START_REF] Kokotov | Tau-functions on spaces of Abelian differentials and higher genus generalization of Ray-Singer formula[END_REF][START_REF] Kokotov | Isomonodromic tau function on the space of admissible covers[END_REF][START_REF] Korotkin | Tau function and moduli of differentials[END_REF][START_REF] Korotkin | Tau function and the Prym class[END_REF]. From now on we assume that all zeros of w are simple i.e. that all k i = 1 in (3.3).

The homology group H 1 ( C, C) can be decomposed into the eigenspaces of the automorphism σ * :

H 1 ( C, C) = n-1 i=0 H k ,
where dimH 0 = 2g. Since all zeroes of w are assumed to be simple, the dimensions of H k for k = 1, . . . , n -1 are independent of k and we have (3.7) dim

H k = (2n + 2)(g -1) , k = 1, . . . , n -1 .
For any two classes s 1 ∈ H l and s 2 ∈ H k we have s 1 •s 2 = 0 unless k+l = n, where • denotes the intersection pairing. The spaces H k and H n-k are, therefore, dual to each other with respect to the standard intersection pairing (the space H 0 can be identified with H 1 (C), and, therefore, it is self-dual). On the other hand, for any q ∈ Ω (k) and s ∈ H l we have s q = 0 unless k = l. In particular, since v = w 1/n ∈ Ω (1) , it can have non-trivial periods only over the cycles representing homology classes in H 1 . Actually, H 1 can be naturally identified with the tangent space to the moduli space M To study the tau function on M

(n) g we will need a variational formula for B(x, y). Consider a fundamental polygon C of C (that is, dissect C along the cuts representing the Torelli marking). Choose a system Γ of non-intersecting cuts that lie within C and connect the first zero x 1 with every other zero of w. Pick a connected component of f -1 ( C \ Γ) ⊂ C and identify it with C \ Γ. On C \ Γ introduce the coordinate

(3.10) z(x) = x x1 v ,
where the path connecting x with x 1 entirely lies in C \ Γ. For a basis {s i }

(2n+2)(g-1) i=1 in H 1 take the dual basis {s * j } (2n+2)(g-1) j=1 in H n-1 , so that s * j • s i = δ ij .
Theorem 3.5. The following variational formula holds for i = 1, . . . , (2n+2)(g-1):

(3.11) ∂ ∂P i B(z(x), z(y)) = 1 2π √ -1n t∈s * i B(z(x), t) B(t, z(y)) v(t) dt .
Proof. Formula (3.11) follows from the variational formula for the Bergman kernel on the stratum H ĝ (n, . . . , n) of the moduli space of holomorphic 1-differentials of genus g = n 2 (g -1) + 1 (Theorem 3 of [START_REF] Kokotov | Tau-functions on spaces of Abelian differentials and higher genus generalization of Ray-Singer formula[END_REF]), in a way similar to Lemma 5 of [START_REF] Korotkin | Tau function and the Prym class[END_REF] and Proposition 3.2 of [START_REF] Bertola | Symplectic geometry of the moduli space of projective structures in homological coordinates[END_REF]. Namely, consider a pair ( C, v) representing a point in the space H ĝ (n, . . . , n). Denote by B(x, y) the canonical bidifferential (the Bergman kernel) on C × C. Then we have the following variational formula for B(x, y) [START_REF] Kokotov | Tau-functions on spaces of Abelian differentials and higher genus generalization of Ray-Singer formula[END_REF]:

(3.12) ∂ B(z(x), z(y)) ∂( si v) = 1 2π √ -1 t∈s * i B(z(x), t) B(t, z(y)) v(t) dt .
Here

{s i } is a canonical basis in H 1 C, {x i } n(2g-2) i=1
, {s * j } is the dual basis in

H 1 C \ {x i } n(2g-2) i=1
(so that s i • s * j = δ ij ), and x 1 , . . . x n(2g-2) are the zeroes of v. Let now the pair ( C, v) be the canonical n-fold cover of (C, w). We require the Torelli markings of C and C be compatible (i.e. each connected component of f -1 (a i ), i = 1, . . . , a g , belongs to the subgroup generated by âj , j = 1, . . . , a ĝ , where a i and âj are the a-cycles on C and C respectively). Then averaging B(x, y) in both variables over the action of the group Z/nZ on C we get the formula

(3.13) B(x, y) = 1 n n-1 l,k=1 (σ * x ) k (σ * y ) l B(x, y)
that can be checked by comparing the singularities of the both sides. Restricting now formula (3.12) from H ĝ (n, . . . , n) to M

(n) g and using (3.13) we get (3.11).

Consider the Schwarzian derivative S

v = v v -3 2 v v 2
of the abelian integral

∂ ∂P j s * i S B -S v v = 1 12π √ -1n s * j s * i B(x, y) B(y, x) v(x)v(y) dxdy
under the transposition of i and j.

Asymptotics of tau function near the boundary of M

(n)

g . Here we compute the asymptotics of τ near generic point of every boundary stratum .

M (n) g \ M (n) g [1] of M (n) g .
Proof. This is analogous to the proof of Lemma 8 of [START_REF] Korotkin | Tau function and the Prym class[END_REF]. Denote by ζ a local coordinate in a small disk U containing the coalescing zeros x 1,2 and no other zeros. Then we can write in U Proof. The asymptotics (3.17) can be derived by computing the asymptotics of all factors in the explicit formula (3.3). Alternatively, using the system of equations (3.14) we see that in the limit t deg → 0 the tau function τ (C, w) behaves like t p deg τ (C 0 , w 0 ) for some power p, where w 0 is a differential with one zero of order two and every other zero being simple. To find p explicitly, we look at the transformation properties of τ , τ 0 and t deg under the rescaling w → δw. The homogeneity degrees κ of τ and κ 0 of τ 0 are given by the formula (3.6), so that

(3.16) w(ζ) = (ζ -ζ(x 1 ))(ζ -ζ(x 2 ))(dζ) n , so that x2 x1 v = ζ(x2) ζ(x1) ((ζ -ζ(x 1 ))(ζ -ζ(x 2 ))) 1/n dζ = const • (ζ(x 1 ) -ζ(x 2 )) (n+2)/n . Since (ζ(x 1 ) -ζ(x 2 ))
κ -κ 0 = 1 6n(n + 1)(n + 2)
On the other hand, the local parameter t deg has the homogeneity degree 2/(n + 2), which gives p = 1 12n(n+1) .

3.3.2. Asymptotics of τ near D 0 . Take two loops a and b on C intersecting transversely at one point; their homology classes we will also denote by a, b ∈ H 1 (C, Z). Let us pinch a to a point, then C degenerates to a nodal curve C 0 that we represent by a smooth curve of genus g -1 with two points (say, x 0 and y 0 ) identified (we assume that all zeros of w remain far from the node).

The holomorphic n-differential w degenerates to a meromorphic n-differential w 0 on C 0 with poles of degree n at x 0 and y 0 on the normalization of C 0 such that the corresponding n-residues differ by (-1) n . Since locally near both x 0 and y 0 the n-differential w behaves like (dζ/ζ) n , no branching over the node occurs.

Because of that the canonical cover C of C degenerates to a nodal curve C 0 with exactly n nodes that can be thought of as n pairs of points x (m) 0 = σ m (x 0 ) and y (m) 0 = σ m (y 0 ) (m = 0, . . . , n -1) on the normalization of C 0 that are pairwise identified (here σ is the covering automorphism of C 0 ). The differential v on C degenerates to a meromorphic differential v 0 on C 0 with simple poles at the preimages of nodal points with residues at x (m) 0 and y (m) 0 that differ by a sign. Choose one of the n simple closed loops in the preimage f -1 (a) ⊂ C. Let us assume that this loop pinches to the first node on C 0 .

Consider the classes α, β ∈ H 1 given by

(3.18) α = n-1 m=0 ρ -m σ -m * a , β = n-1 m=0 ρ -m σ -m * b
where σ is the covering automorphism of C. Introduce the homological coordinates P α = α v and P β = β v associated with α and β. We have α v = n a v and

β v = n b v.
Since all P j are independent local coordinates, we can choose such a path towards D 0 that all coordinates P i remain constant under the degeneration of C to C 0 while P β → ∞. Then a local coordinate on M (n) g transverse to D 0 in a tubular neighborhood of D 0 can be chosen as (3.19) t 0 = e 2πiP β /Pα (notice that Im(P β /P α ) > 0 near D 0 ).

To show this one can use Fay's "plumbing construction", cf. p. 52 of [START_REF] Fay | Kernel functions, analytic torsion, and moduli spaces[END_REF]. Roughly speaking, the degenerating curve C can be obtained from the nodal curve C 0 by cutting off a neighborhood of each node and connecting the boundaries with long thin tubes. More precisely, in terms of a local coordinate t near D 0 the differential v can be written as Pα 2πi dx √ x 2 -t + . . . . Then the coordinate t 0 (3.19) is given by t(const + o(1)).

Let ω x,y be the abelian differential of the 3rd kind on C 0 with simple poles at points x and y of residues +1 and -1 respectively, normalized with respect to some canonical basis {a i , b i } in H 1 ( C 0 , Z). Since v 0 ∈ H 1 , it can be written as

v 0 = P α 2π √ -1 n-1 j=0
ρ m ω σ m (x0), σ m (y0) + holomorphic terms .

In the limit t 0 → 0 the bidifferential B(x, y) on C × C tends to the meromorphic bidifferential B 0 (x, y) on C 0 × C 0 with the same properties. We denote by B 0 (x, y) the pullback of B 0 (x, y) to C 0 × C 0 .

To find the asymptotics of the tau function τ as t 0 → 0 (i.e. P β → ∞), consider the equation

(3.20) ∂ log τ ∂P β = - 1 12π √ -1n β * S B -S v v -→ t0→0 - 1 6n Res x0 S B0 -S v0 v 0 ,
where ) denote the preimages of the node in C 1 (resp. C 2 ) that are cyclically ordered relative to the covering maps σ i : for each m = 0, . . . , n -1 (since the zeroes of both w 1 and w 2 stay away from the node of C 0 , no branching over x 0 ∈ C 1 and y 0 ∈ C 2 can occur).

β * = 1 n n-1 m=0 ρ m σ m * a ∈ H n-1 is the class dual to β ∈ H 1 . To compute the residue, choose a local coordinate ζ near x 0 on C 0 such that S B0 = 0. Then we have v 0 = Pα 2π √ -1n dζ ζ + O(1) and S v0 = v 0 v0 -1 2 v 0 v0 2 = 1 2ζ 2 + O(1) as ζ → 0, so that S v0 v 0 = π √ -1n P α dζ ζ + O (1). Therefore, (3.20) implies (3.21)
C i → C i . The canonical cover C 0 of the nodal curve C 0 is obtained from C
Define the differentials v i (i = 1, 2) on C i by putting v n i = f * i w i and choosing the roots of unity in such a way that Res| x (m) 0 v 1 = -Res| y (m) 0 v 2 (note that v 1 and v 2 have first order poles at the n preimages of the node). Moreover, applying m times the cover map σ 0 , we see that Res| x (m)

0 v 1 = ρ -m Res| x (0) 0 v 1 (resp. Res| y (m) 0 v 2 = ρ -m Res| y (0) 0 v 2 ).
The preimage f -1 (γ) ⊂ C of the loop γ on C is the disjoint union of n loops γ m , m = 0, . . . , n -1 (we enumerate them in such a way that γ m+1 = σ(γ m ), assuming that γ n = γ 0 ). Note that the union of γ m , m = 0, . . . , n -1, is homologically trivial on C. Consider also a simple loop η 0 on C such that γ 0 • η 0 = 1, γ 1 • η 0 = -1, and γ k • η 0 = 0 for k = 2, . . . , n -1, where • stands for the intersection pairing of 1-cycles on C.

Introduce the loops η m = σ m (η 0 ), m = 1, . . . , n -1, and consider the cycles

(3.23) α = n-1 m=1 (ρ -m -1)γ m , β = 1 ρ -1 n-1 m=1 (1 -ρ -m )η m
(for homology classes in H 1 ( C, Z) represented by γ m and η m we use the same notation). Clearly, α, β ∈ H 1 , and the class β * ∈ H n-1 dual to β is given by (3.24)

β * = 1 n n-1 m=1 (ρ m -1)γ m .
Consider the homological coordinates (3.25)

P α = α v = n γ0 v , P β = β v = n 1 -ρ η0 v .
By Fay's plumbing construction [START_REF] Fay | Kernel functions, analytic torsion, and moduli spaces[END_REF], the degenerating covering curve C can be represented by C 0 with nodes replaced by n long thin cylinders in the flat metric on C. Therefore, while C degenerates to C 0 all homological coordinates except P β remain finite. Similarly to the case of D 0 , (1/n)P α equals to the integral of v along the vanishing cycle on C. Therefore, in complete analogy to the case of D 0 the local parameter transverse to D j ⊂ M

(n) g can be chosen as To find the asymptotics of τ when t j → 0, we use Eq. (3.14) once again:

∂ log τ ∂P β = - 1 12π √ -1n β * S B -S v v (3.27) = - 1 12π √ -1n γ0 S B -S v v -→ - 1 6n Res| x (0) 0 S B0 -S v0 v 0 .
Pick a coordinate ζ near x We denote by Z i the closure of Z i . This is a closed substack of P M (n) g,1 of pure codimension i for 1 ≤ i < N = (2g -2)n, while for i = N it has two components of codimensions N -1 and N respectively, cf Section 1.2.

Let

π : P M (n) g,1 → P M (n) g
be the forgetful map of the marked point. Then it is easy to see that the image of Z 2 under π is the divisor D deg . This statement takes into account the multiplicities of the components of D deg . Indeed, if (g, n) = (2, 2) then the restriction of π to Z 2 is of degree 1 and if (g, n) = (2, 2) then π is of degree one onto P M This line bundle has a natural section s 1 : (C, w) → w(x 1 ).

In other words, s 1 is the evaluation of w at the marked point. The class of the vanishing locus of s 1 is given by the first Chern class of the line bundle:

{s 1 = 0} = c 1 O(1) ⊗ L ⊗n 1 = -ψ + nψ 1 .
It is easy to see that Z 1 ⊂ {s 1 = 0}. In the next section we will show that the vanishing locus has no other components and that the vanishing order of s 1 along Z 1 is equal to 1. It will follow that [Z 1 ] = -ψ + nψ 1 . Now we restrict to Z 1 and study the line bundle O(1) ⊗ L ⊗n+1

1

. This line bundle has a natural section s 2 : (C, w) → w (x 1 ).

In other words, assuming that w vanishes at x 1 , the section s 2 assigns to w its derivative at x 1 . It is easy to see that Z 2 is a component of the vanishing locus {s 2 = 0}. In the next section we will show that the vanishing locus has no other components and that the vanishing order of s 2 along Z 2 is equal to 1. Thus we have [Z 2 ] = (-ψ + (n + 1)ψ 1 )[Z 1 ] = (-ψ + nψ 1 )(-ψ + (n + 1)ψ 1 ).

(This computation seems standard, but we were not able to find a proper reference in the literature.) Recall that δ deg is the push-forward by π of this expression. To compute this push-forward we use

• ψ is a pull-back under π; • π * ψ 1 = 2g -2; • π * ψ 2 1 = κ 1 = 12λ 1 - [g/2] i=1 δ i .
The last equality is the well-known Mumford's formula. Applying these equalities we get

δ deg = π * ((-ψ + nψ 1 )(-ψ + (n + 1)ψ 1 )) = n(n + 1)π * (ψ 2 1 ) -(2n + 1)π * (ψ 1 )ψ = 12n(n + 1)λ 1 -n(n + 1) [g/2] i=1 δ i -(2n + 1)(2g -2)ψ.
This coincides with the expression of Theorem 3.8.

In order to complete the proof of Theorem 3.8, it remains to prove that the vanishing locus of s 1 (respectively s 2 ) is exactly Z 1 (respectively Z 2 ) and that the vanishing order of s 1 and s 2 is 1. Vanishing loci of s 1 and s 2 . Let W be an irreducible divisor of P M (n) g,1 in the vanishing locus of s 1 . Let k be the number of nodes of the curve represented by a generic point of W . The vanishing locus of s 1 is of codimension 1 in P M (n) g,1 thus k = 0 or 1. We investigate both cases.

• Let k = 0. Then a dense subset of W is contained in Z 1 and thus W is a component of Z 1 . • Let k = 1. Then the divisor W is contained in ν-1 (D) for some irreducible boundary divisor D of the moduli space of stable curves with one marked point (we recall that ν : P M

g,1 → M g,1 is the forgetful map). Since the D is irreducible, and ν is the projectivization of a vector bundle, we necessarily have W = ν-1 (D). However there exists an n-differential in ν-1 (D) which is not identically zero on the component containing the marked point. Therefore, there exists a point in ν-1 (D) which is not in the vanishing locus of s 1 . Thus the case k = 1 does not occur. To study the vanishing locus of s 2 we follow the same strategy. First we can check by dimension count that no irreducible component of Z 1 is in the zero locus of s 2 . Now let W be an irreducible divisor in the vanishing locus of s 2 and let k be the number of nodes of the curve represented by a generic point of W . We have now 3 cases to study: k = 0, 1 and 2.

• Let k = 0. Then a dense subset of W is contained in Z 2 and thus W is a component of Z 2 . • Let k = 2. Then W = ν-1 (D) where D is a codimension 2 boundary stratum of M g,1 . As above ν-1 (D) is not contained in the vanishing locus of s 2 . Thus the case k = 2 cannot occur.

• Let k = 1. Then W is a co-dimension 1 locus inside ν-1 (D) for a boundary divisor D of M g,1 .
The generic curve has two components of genera g and g -g with 1 ≤ g ≤ g -1. We assume that the marked point is carried by the component of genus g . The rank of the bundle of n-differentials with a pole of order at most n at the node is n(2g -2 + 1) > 1. Thus the divisor D is not contained in the locus of differentials that vanish identically on this component. Thus D is not contained in the vanishing locus of s 2 . We conclude that {s i = 0} = Z i for i = 1 and 2.

Vanishing order of s 1 . Let y 0 = (C 0 , w 0 , x 0 ) be a point in Z 1 . We recall that

P M (n) g,1 → P M (n) g
is isomorphic to the universal curve. Thus a neighborhood of y 0 in Z 1 is given by U × ∆ where U is a neighborhood of (C 0 , w 0 ) in P M (n) g [START_REF] Babelon | Introduction to classical integrable systems[END_REF] and ζ ∈ ∆ is the distinguished parameter around x 0 in C 0 (cf Section 2.1). Let (C, w, x) be an n-differential in U ×∆. In coordinates (u, ζ) ∈ U ×∆, the differential w is given by w = ζdζ n , the locus Z 1 is {ζ = 0} and the section s 1 is given by s 1 (u, ζ) = ζ. Therefore the vanishing order of s 1 along Z 1 is 1.

Vanishing order of s 2 . Let y 0 = (C 0 , w 0 , x 0 ) be a point in Z 2 . A neighborhood of y 0 in P M (n) g,1 is now given by U × ∆ × ∆ where U is a neighborhood of y 0 in Z 2 and ∆ and ∆ are disks in the complex plane centered at 0 and parametrized by ζ and a such that:

w = (ζ 2 + a)dζ n .
With the parameters (u, ζ, a) ∈ U × ∆ × ∆ , the locus Z 1 is defined by ζ 2 + a = 0. Moreover with these parameters, the section s 2 is given by s 2 (u, ζ, a) = a. Thus the vanishing order of s 2 along Z 2 is again 1.

Prym-Tyurin differentials on C and holomorphic n-differentials on C

Here we relate Prym-Tyurin vector bundles to vector bundles of holomorphic k-differentials on the base Riemann surface C. We use this relation to finish the proof of Theorem 1.12.

We also prove that the Prym-Tyurin bundle is not a pullback from P M

(n) g in general.

5.1. Prym-Tyurin bundles and n-differentials. Consider two vector bundles Λ (k) and ν * Ω (n-k+1) g over X(g, n). The fiber of Λ (k) is the kth eigenspace in the space of abelian differentials on C. The fiber of Ω n-k+1 is the space of (n -k + 1)differentials on C. There are natural morphisms:

(5.1) Φ 0 :

Λ (0) → ν * Ω g , Φ k : Λ (k) ⊗ T ⊗(n-k) → ν * Ω (n-k+1) g for 1 ≤ k ≤ n.
Indeed, let (C, w) be a point in U \D deg and let q be a differential in the fiber of Λ (k) . The n -k + 1 differential qv n-k is invariant under the action of the automorphism group of the cover, thus qv n-k is the pull-back of n -k + 1 differential on C. For k = 0 the differential q is already invariant under the action of the automorphism group of the cover, so there is no need to multiply it by a power of v. Proof. On the locus where Φ k is an isomorphism we have ).

Thus the difference between λ (k) P T and λ n-k+1 -g-1 n (n -k)(2n -2k + 1)ψ is an effective divisor defined as the vanishing locus of det Φ k .

In Section 2.2, we have described a parametrization of the cyclic covers along D deg . We use here this parametrization to prove the following Lemma. This corollary together with Theorem 3.8 and Formula (1.6) completes the proof of Theorem 1.12.

Remark 5.6. Note the difference of a factor 2 between the vanishing order of det Φ k and the contribution of δ deg in λ (k) P T . This is due to the fact that V → U is of degree 1/2 along D deg .

The proof of Lemma 5.4 will occupy the two following sections. We consider separately the cases of even n and odd n. zero of w 0 . We define q

(k) 0 = (ζ 1 -ζ 2 ) -1+2k/n • f * (q (k) 0 ) v k
. and we study the asymptotic behavior of Φ k (q (k) 0 ⊗ v n-k ) along D deg . As in the odd case, the differential q (k) 0 is a non vanishing section of Λ (k) and Φ k (q

(k) 0 ⊗ v n-k ) = (ζ 1 -ζ 2 ) -1+2k/n • q(k)
0 by construction. Therefore the vanishing order of det(Φ k ) is given by 1 -2k n for k = 1, . . . , m -1. g . In order to define the Prym-Tyurin classes we have constructed the space of admissible differentials X(g, n) (see 1.5). Indeed, the Prym-Tyurin bundles are naturally defined over X(g, n). The following theorem explains the necessity of the introduction of the space X(g, n).

Theorem 5.8. Let g > 2 and k > 0. There exists no vector bundle Λ → P M (n) g such that Λ (k) = diff * Λ(k) , where diff : X(g, n) → P M (n) g is the forgetful map.

Proof. Suppose that there exists Λ → P M (n) g such that Λ (k) = diff * Λ(k) . Then in particular λ k P T = c 1 ( Λ(k ) and thus c 1 (Λ k ) = diff * λ k P T . We will prove that this equality does not hold for g > 2 and k > 0.

Let m > 2 be such that gcd(m, n) = 1. Let µ be the partition of n(2g -2) given by (m, 1, . . . , 1). We denote by M

1 . 4 .

 14 Moduli space of n-differentials 1.2. Stratification of P M (n) g 1.3. First definition of Prym-Tyurin classes 1.4. Admissible covers and Prym-Tyurin bundles 1.5. Space of admissible differentials 1.6. Statement of the results 1.7. Strategy of the proof 2. The space of admissible n-differentials 2.1. Distinguished local coordinates on a cyclic cover 2.2. Extension of the Prym-Tyurin bundles to codimension 1 loci 3. Bergman tau function and Hodge class on P M (n) g 3.1. Bergman tau function on strata of n-differentials 3.2. Homological coordinates and variational formulas for the tau function 3.3. Asymptotics of tau function near the boundary of M (n) g 3.4. Hodge class on P M (n) g An alternative computation of δ deg 5. Prym-Tyurin differentials on C and holomorphic n-differentials on C 5.1. Prym-Tyurin bundles and n-differentials 5.2. Odd n = 2m + 1 5.3. Even n = 2m 5.4. Obstruction to the extension of the Prym bundles to M

  dual classes δ 0 , . . . , δ [g/2] ∈ Pic(M g ) of the boundary divisors D 0 , D 1 , . . . , D [g/2] ⊂ M g .

g

  [k] is empty or it has pure dimension 2g -3 + m. • If all k i 's are divisible by n then P M (n) g [k] has components of dimension 2g -2 + m. A differential (C, w) lies in a component of dimension 2g -2 + n if and only if w is the nth power of a holomorphic differential. The stratum P M (n) g [k] may also have irreducible components of dimension 2g -3 + m composed of n-differentials that are not nth powers.

2 .

 2 We denote by δ deg the cohomology class that is Poincaré dual of D deg . Remark 1.4. Heuristically, D deg is the divisor of n-differentials with a double zero, and for (g

Remark 1 . 7 .

 17 By abuse of notation we denote by λ (k) P T = det Λ (k) both determinant line bundle and its class in the Picard group. Let us mention the following fact: for any (C, w) ∈ M (n) g

1 . 5 .

 15 n g and P M (n) g and use the push-forward of a pull-back. Space of admissible differentials. Definition 1.8. Let I :

Definition 1 . 11 . 6 .

 1116 The Prym-Tyurin class λ diff * c 1 (Λ (k) ). 1.Statement of the results. The main result obtained in this paper is the expression for the Prym-Tyurin classes and the class δ deg of Definition 1.3 in the basis (λ, ψ, δ i ).

  |V * . Thus the morphism diff * and diff * (from Pic(X(g, n)) to Pic(P M (n) g )) constructed in Section 1.5 are equal. The same stands for diff * and diff * thus finishing the proof. 2.1. Distinguished local coordinates on a cyclic cover. Consider a curve C endowed with an n-differential w with simple zeros. Let f : C → C be the associated cover and v = w 1/n be the canonical abelian differential on C. Denote by x i ∈ C, i = 1, . . . , N = 2n(g -1), the branch points of C → C, which coincide with the zeros of w. Denote by xi the unique preimage of x i in C. Here we describe a specific parametrization of the covering curve C near the branching points of f : C → C. It is easy to see that all zeros of the holomorphic 1-form v are situated at the ramification points xi and have multiplicity n. In other words, (2.1)

Lemma 2 . 3 (

 23 Property 2). The stack U is an open substack of P M (n) g

  (n) g[k] for all k except (1, . . . , 1) and (2, 1 . . . , 1). Thus U is an open substack in P M (n) g and by dimension count its complement is of codimension at least 2.

Lemma 2 . 4 (

 24 Property 3). The restriction of diff : V → U induces an isomorphism on the underlying schemes. Moreover the map of stacks diff : V → U is of degree one over U \ D deg and of degree 1/2 over D deg . Remark 2.5. The reader should keep in mind the model case of the quotient Spec(C[x])/(Z/2Z

Figure 1 . 3 .

 13 Figure 1. Examples of degeneracy of canonical n-covers (before colliding the zeros): on the left n = 4 and on the right n = 3.

gProposition 3 . 2 .

 32 [k]. Other properties of τ follow below. Under the change of Torelli marking on C 2g, Z) , the tau function (3.3) transforms as follows:

6 S

 6 a basis {s i } (2n+2)(g-1) i=1in H 1 we introduce local coordinates P i on M homological coordinates (cf. Corollary 2.3 of[START_REF] Bainbridge | Strata of k-differentials[END_REF]).Choose a Torelli marking on C and define the associated canonical bimeromorphic differential B(x, y) = d x d y log E(x, y) on C. The bidifferential B (also called the Bergman kernel) is symmetric with a second order pole with biresidue 1 on the diagonal x = y and vanishing a-periods with respect to the both of its arguments. Denote by B(x, y) = B(f (x), f (y))df (x)df (y) the pullback of B(x, y) to C × C. The bidifferential B(x, y) has the following expansion near x = y: B (ζ(x)) + . . . dζ(x)dζ(y) , where ζ is a local parameter and B is the Bergman projective connection (lifted to C).

x x0 vS

  with respect to the local coordinate z on C introduced earlier. For the holomorphic 1-differential v, S v is a meromorphic projective connection on C, so that the difference S B -S v is a meromorphic quadratic differential.The tau function τ = τ (C, w) satisfies the following system of differential equations with respect to the homological coordinates P i on P M B -S v v (equations (3.14) for the tau function τ (C, w) are restrictions of the equations for the Bergman tau function on the stratum H g (n, . . . , n) of the moduli space of Abelian differentials[START_REF] Kokotov | Tau-functions on spaces of Abelian differentials and higher genus generalization of Ray-Singer formula[END_REF] to the space M (n) g ). The compatibility of the system (3.14), namely, the fact that ∂ ∂Pj ∂ ∂Pi log τ = ∂ ∂Pi ∂ ∂Pj log τ follows from the invariance of the expression

3. 3 . 1 .

 31 Coalescing simple zeros of w: divisor D deg . Lemma 3.6. Let x 1 and x 2 be two zeros of w coalescing at D deg . Then a transverse local coordinate on M (n) g in a tubular neighborhood of D deg is given by

Lemma 3 . 7 .

 37 The tau function τ (C, w) has the following asymptotics near D deg :(3.17) τ (C, w) = t 1 12n(n+1) deg τ 0 (C 0 , w 0 )(1 + o(1)) ,where (C 0 , w 0 ) ∈ D deg and τ 0 (C 0 , w 0 ) is the tau function (3.3) on the moduli space M (n) g [2, 1, . . . , 1].

3 .

 3 Asymptotics of τ near D j . Contracting a homologically trivial simple loop γ on C we get a reducible nodal curve C 0 that splits into two irreducible components C 1 and C 2 of genera g 1 = j and g 2 = g -j respectively, j = 1, . . . , [g/2]. Denote by x 0 ∈ C 1 and y 0 ∈ C 2 the intersection point of C 1 and C 2 (the node of C 0 ). The holomorphic n-differential w on C degenerates to a pair of meromorphic ndifferentials w 1 and w 2 on C 1 and C 2 respectively, with poles of order n at x 0 ∈ C 1 and y 0 ∈ C 2 whose n-residues differ by (-1) n (we assume that under generation the zeroes of w stay away from the vanishing cycle γ).Denote by f i : C i → C i the canonical n-fold cover of the curve C i (i = 1, 2), and let x

( 3 .

 3 26)t j = e 2π √ -1P β /Pα .

0 1 )

 1 such that S B0 = 0, then v 0 = Pα as t j → 0. Therefore, (3.27) implies ∂ log τ ∂P β tj =0

3. 4 . 12 [g/2] j=0 δ j . 4 .

 4124 Hodge class on P M (n) g . A straightforward combination of Theorem 3.4 with asymptotic formulas (3.17), (3.22) and (3.28) yields Theorem 3.8. (Formula (1.3) of Theorem 1.12) The Hodge class λ on M (n) g is a linear combination of the tautological class ψ and the classes of boundary divisors D deg and D j , j = 0, 1, . . . , [g/2], as follows: (3.29) λ = (g -1)(2n + 1) 6n(n + 1) ψ + 1 12n(n + 1) δ deg + 1 An alternative computation of δ deg Let g, n ≥ 2. In order to compute the class δ deg in Pic(P M (n) g ) we begin with marking one point on C, i.e. we study the space P M (n) g,1 . In P M (n) g,1 we define the loci Z i = {(C, x 1 , w) | C is smooth and x 1 is a zero of w of order at least i}.

1 Hom L, L ⊗n 1 .

 11 (n) g (2, 1, 1) and two onto P M(n) g (2, 2). Therefore π * [Z 2 ] = δ deg . Thus to find an expression of δ deg it suffices to compute the class [Z 2 ] ∈ A 2 (P M (n) g,1 ). Computation of [Z 2 ]. Let L 1 → M g,1be the line bundle whose fiber is the cotangent line to the curve C at x 1 , and put ψ 1 = c 1 (L 1 ). Consider the following line bundle over P M

Lemma 5 . 1 .

 51 The map Φ k is an isomorphism over V \ D deg .Proof. The maps Φ k for 0 ≤ k ≤ n -1 are injective because v does not vanish identically on any component of the nodal curve C. They are bijective because the sum of ranks of the Prym-Tyurin bundles Λ (k) for 0 ≤ k ≤ n -1 is equal to the sum of ranks of the vector bundles Ω n-k+1 . Indeed,n-1 k=0 rk Λ (k) = rk Ω g = n 2 (g -12k + 1)(g -1) = n 2 (g -1) + 1.Corollary 5.2. The rank of the Prym-Tyurin bundle is g for k = 0 and (2n -2k + 1)(g -1) for 1 ≤ k ≤ n -1.

Corollary 5 . 3 .

 53 In Pic(P M (n) g \ D deg ) we have (5.2) λ (k) P T = λ n-k+1 -g -1 n (n -k)(2n -2k + 1)ψ for 1 ≤ k ≤ n -1.

= c 1

 1 (Λ (k) ) = c 1 (ν * Ω k ⊗ T ⊗-(n-k) ) = λ n-k+1 -(g -1)(n -k)(2n -2k + 1)c 1 (T ) = λ n-k+1 -g -1 n (n -k)(2n -2k + 1)ψ,where the last equality is due to T ⊗n = L.The locus where Φ k is an isomorphism contains P M(n)g \ D deg up to codimension 2 substacks that are immaterial for the first Chern class computations.To study the extension of the formula (5.2) to P M (n) g we need to study the behavior of Φ k along the boundary divisor D deg . The determinant of Φ k is a global section of det(Λ (k) ⊗ T n-k ) -1 ⊗ det(ν * Ω (n-k+1) g

Lemma 5 . 4 .

 54 If [(n -1)/2] + 1 ≤ k ≤ n -1 or k = 0, then the morphism Φ k is an isomorphism of vector bundles over V ⊂ X(g, n). Otherwise, det(Φ k ) vanishes along D deg with order (1 -2k n ). This lemma implies the following Corollary 5.5. The following relations between Prym-Tyurin class λ (k) P T , the class ψ = c 1 (L) and the pullback of class λ n-k+1 from M g to P M λ n-k+1 -g -1 n (n -k)(2n -2k + 1λ n-k+1 -g -1 n (n -k)(2n -2k + 1)ψ , (5.4) [(n -1)/2] + 1 ≤ k ≤ n -1 .

5. 2 .

 2 Odd n = 2m + 1.

5. 4 .

 4 Obstruction to the extension of the Prym bundles to M (n)

  denote by D m the preimage of M (n) g[m] in X(g, n) under diff. The locus D m is a divisor whose generic points are elements (C, w, x i , f : C → C) ∈ X(g, n) such that:• the curve C is a nodal curve with two components: a component C 2 isomorphic to C with n(2g -2) -m marked points intersecting a rational component carrying m marked points in one point that is located at the zero of order m of w on C 2 . • the n-differential w is identically zero on C 1 and has profile µ on C 1 ;• the cover curve C → C has two component C 2 and C 1 . The component C 2 determined by the w as in Section 2.2 and C 1 → C 1 is the unique n-sheeted ramified cover maximally ramified at the marked points and the node. Moreover the canonical root v of f * w vanishes identically on C 1 and has a zero of order m + n -1 at the preimage of the zero of order m. Thus the morphism Φ k : Λ (k) ⊗T ⊗(n-k) → ν * Ω (n-k+1) g has a non-empty co-kernel along D m for m large enough. Therefore the line bundle det(Λ (k) ) ⊗ diff * det Λ(k)∨ has a global section which vanishes along divisors contained in X(g, n) \ V . Thus Λ (k) = diff * Λ(k) .

  1 and x 2 are given by ζ 1 and ζ 2 , respectively. To fix ζ uniquely we can define it by:

2Z)

where ∆ is a disk in C centered at zero. Indeed, suppose first that the two colliding zeros of a differential (C, w) are labeled x 1 and x 2 . Let ζ be a local parameter of C such that the positions of x

  2 is independent of labeling of zeroes, t deg is a coordinate transverse to D deg . Expressing t deg in terms of

Acknowledgements. DK and PZ acknowledge the hospitality of the Max-Planck-Institut für Mathematik in Bonn where this work began in 2011, and DK also thanks the Max-Planck Institute for Gravitational Physics (Albert Einstein Institute) in Golm where this work was continued. Research of DK was supported in part by the Natural Sciences and Engineering Research Council of Canada grant RGPIN/3827-2015, by the FQRNT grant "Matrices Aléatoires, Processus Stochastiques et Systémes Intégrables" (2013PR 166790), by the Alexander von Humboldt Stiftung and by the Chebyshev Laboratory of St. Petersburg State University. Research of Section 3 was supported by the Russian Science Foundation grant 16-11-10039. The authors are very grateful to Dimitri Zvonkine for his helpful advice and comments. AS is also grateful to Dawei Chen, Charles Fougeron, Martin Möller and Anton Zorich for very fruitful conversations.

5.2.1. Kernel and cokernel of Φ k . Let (C 0 , w 0 ) be a point in D deg and f : C 0 → C 0 be the associated admissible cover. We recall that C 0 is a curve with two components intersecting in one point. The two components C 1 and C 2 are of genera g 1 = m and g 2 = g -m (see Section 2.1). We have denoted by w 1 the meromorphic n-differential on Denote the fiber of kth Prym-Tyurin vector bundle Λ (k) over the point

where Ω (k) 1

is the space of holomorphic differentials on C 1 which get multiplied by ρ k under the action of the automorphism (y,

is the analogous space of holomorphic differentials on C 2 .

The canonical differential v vanishes identically on C 1 . Thus the kernel of Φ k contains the space Ω (k) 1 ⊗T ⊗n-k . On the other hand, the restriction of the morphism Φ k to the linear subspace Ω

over a generic point of D deg .

We have dim Ω (k) 1

= 1 for k = 1, . . . , m. These one-dimensional spaces are generated by the holomorphic differentials on C 1 given by q

We can also describe the images of Φ k . For

). However, for k = 1, . . . , m the image of Ω (k)

2 ⊗T ⊗n-k is the space of holomorphic n-k+1 differentials vanishing at x 0 .

We have proved the following Lemma 5.7. The kernel of Φ k over a generic point of D deg is the vector bundle

the image of Φ k is the vector bundle whose fibers are the H 0 (C, ω n+k-1 (-x 0 )) where x 0 is the unique zero of order 2.

Therefore the first part of Lemma 5.4 and Formula (5.4) are valid for odd n.

Description of

We have seen that the kernel and cokernel of Φ k are of dimension 1. Let (C 0 , w 0 ) be a generic point in D deg . Let W be an open neighborhood of (C 0 , w 0 ) in X(g, n) with a non-vanishing section q

thus the vanishing order of det Φ k is equal to the vanishing order of Φ k (q (k) 0 ⊗ v ⊗n-k ) along D deg . Therefore, we will construct such a local section q (k) 0 of Λ (k) and study the asymptotic behavior of q

0 be a non-vanishing section of ν * (Ω (n-k+1) ) over W such that: for all (C, w) ∈ D deg , q (k) 0 (C, w) is a differential that does not vanish at the double zero of w. Up to a choice of a smaller W , such a section exists. We label the coalescing zeros by x 1 and x 2 . We chose the parameter of the curve ζ such that position of x 1 and x 2 are ζ 1 and ζ 2 and w = (ζ -ζ 1 )(ζ -ζ 2 )(dζ) n (see Section 2.2). We define

well defined, it is the integral of v between x1 and x2 (see Lemma 3.6).

Over W \ D deg , the differential q

is obviously a non-vanishing section of ν * (Λ (k) ). Besides, at a generic point of D deg the differential q (k) 0 vanishes identically on C 2 because of the factor (ζ 1 -ζ 2 ) -1+2k/n . To compute the limit of q (k) 0 on C 1 , we remark that q (k) 0 is given by n . This finishes the proof of Lemma 5.4 for odd n.

5.3.

Even n = 2m. The proof of Lemma 5.4 is essentially identical to the odd case. The difference between the odd and the even case is in the topolocial type of the degenerate cover and on the orders of singularities of the limit differentials. Thus we only need to transpose correctly the notation from the odd case taking into account this differences in orders of singularities.

Let (C 0 , w 0 ) be a generic point in D deg . Now we have g 1 = m -1 and g 2 = g -m and the two components intersect in two points. Let Ω k 0 be the PT bundle over (C 0 , w 0 ) and let Ω (k) i be the subspace of Ω k 0 of holomorphic differentials supported on C i . We still have the decomposition:

for definition of the parameters).

For k = m, the space Ω (k) 0 contains a 1-dimension subspace of differentials of a third kind. These are differentials with simple poles at the nodal points x Similarly to the case of odd n, for k = 1, . . . , m -1, let q(k) 0 be a section of ν * (Ω (n-k+1) ) over a neighborhood of (C 0 , w 0 ) that does not vanish at the double


Aleksandr Petrusev
email: aleksandr.petrusev@grenoble-inp.fr

Rémy Rigo-Mariani

Vincent Debusschere

Patrick Reignier

Nouredine Hadjsaid

AUTOMATIC DETECTION OF DISTRIBUTED SOLAR GENERATION BASED ON EXOGENOUS INFORMATION

Keywords: ARTIFICIAL INTELLIGENCE, AUTOMATIC PV DETECTION, NEURAL NETWORK

In recent years, the importance of PV generation data for distribution system operations has increased. However, there are still a lot of behind-the-meter solar installations that are not registered by the system operator and are not monitored. This "hidden" generation, therefore, increases the difficulty to operate securely and efficiently the distribution grid. This paper introduces a tool for the automatic detection of such "hidden" behind-the-meter solar generation. It is designed to discriminate the nodes with and without PV generation and is aimed at a high accuracy. The tool consists of a neural network coupled with an analytical classification algorithm, which considers an exogenous information (i.e. node consumption and temperature data). Open-access data about consumption and solar radiation were used to simulate the electrical grid and validate the proposed approach. The implemented solution was tested across all the nodes of the grid and its sensitivity has been analysed with regard to the level of PV penetration and period of observation. The tool is able to recognize the nodes with a new PV installation with an accuracy of up to 100%, depending on the exogenous conditions.

Introduction

Distributed renewable energy, and particularly photovoltaic (PV), showed rapid growth over the past two decades. For this reason, information on PV generation becomes one of the key aspects to perform operations such as voltage management, reconfiguration or state estimation in distribution systems. However, a large number of small household-owned PV plants do not necessarily have connection agreement with the system operator and are therefore not monitored. That generation is then "hidden" for the system operator. It incurs additional uncertainty in the net charge measurement and forecasting, which may affect the quality of the distribution grid operation. Thus, a relevant contribution to improve electrical grid management performance is to increase the system observability with a tool that detects nodes with PV production, based on net charge data. There are several proposed approaches for detecting PV generation in distribution grids, but they often require specific data that the system operator may not have access to, such as solar radiation data. One of the approaches, which identifies customers with PV generation by using data about net energy consumption from smart meters, is introduced in [START_REF] Donaldson | Effective solar prosumer identification using net smart meter data[END_REF]. The authors propose a dimensionality reduction method combined with a classification algorithm, more effective than K-means, to reduce the amount of data needed to accurately identify solar prosumers to a single data point (by taking the minimum, average, and maximum consumption for the entire year). The classification method is based on agglomerative clustering and self-organizing maps (an artificial neural network-based clustering technique). One limitation of the method is that the clustering of consumers into two groups (with and without PV) is carried out based on whether the peak consumption of a consumer is below the average of all customers, without taking into account data on the historical consumption of this consumer in previous years. Thus, the performances and accuracy are strongly affected by the diversity of the considered consumption load profiles. Another algorithm proposes the automatic detection of PV panels by using very high-resolution color satellite imagery (0.3 meters per pixel) [START_REF] Malof | Automatic detection of solar photovoltaic arrays in high resolution aerial imagery[END_REF]. A Random Forest classification machine learning technique detects the presence of PV on an image. The need to use highresolution satellite images along with labeled training data is the main disadvantage of this algorithm. An approach for solar prosumer identification, which uses change-point to detect abnormal energy consumption behaviors (such as not monitored PV generation), is presented in [START_REF] Zhang | A data-driven approach for detection and estimation of residential PV installations[END_REF]. Various abnormalities can cause changepoints in customer load. Therefore, the presence of the not monitored PV generation is further verified through a statistical inference known as permutation test with Spearman's rank coefficient. Nevertheless, this approach is unable to detect PV installations until after the rolling window length is completed (up to 14 days lengths were evaluated), and additionally needed a cloud cover index.

Another approach is a distributed photovoltaic systems capacity estimator [START_REF] Wang | A distributed PV system capacity estimation approach based on support vector machine with customer net load curve features[END_REF]. The algorithm determines whether a customer has a PV, by using Support Vector Machines (SVM). Four weather status-driven features are extracted for SVM, they describe the difference of net charge profiles between customers with and without PV -the ratio of total electricity consumption, concave shape index, concavity degree and load ramping rate. This method needs the PV power output of the known customers, which is its main drawback. Finally, authors in [START_REF] Michaelangelo | Disaggregating solar generation behind individual meters in real time[END_REF] present a method for detecting and disaggregating behind-the-meter solar generation using weather data, advanced metering infrastructure, substation monitoring and generation monitoring from selected nearby PV systems. Similarly, the needs of that method concern power from other nearby PV. The above methods rely on various types of data such as satellite imagery, solar radiation data, data from other PV stations or detailed weather data to be operational. The method proposed in this paper, on the contrary, does not require such data and does not contain any detailed modelling. It strictly relies on smart metering and temperature data. In doing so, the proposed approach avoids user privacy issues that have not been addressed previously and only uses aggregated consumption data at the feeder level, as presented below.

Machine learning techniques as neural networks (NN) could provide a large number of functions in the energy field [START_REF] Ali | State-of-the-Art Artificial Intelligence Techniques for Distributed Smart Grids: A Review[END_REF], the most popular being forecasting and disaggregation. To the best of our knowledge, a PV generation detection method that combines the trade-offs between data limitation, simplicity of implementation and usable results has not been extensively covered. In this paper, the approach of time series forecasting based on NN is applied to a classification task. The proposed method combines a conventional Multi-Layer Perceptron (MLP) together with an analytical classification algorithm. The paper is organized as follows: Section II describes the data and simulation environment for the experiment and presents the method. The obtained results and sensitivity analysis of this method are given in Section III. Finaly, conclusions are drawn in Section IV.

Methodology

Experimental setup

The Pandapower tool was used to simulate the electrical distribution grid [START_REF]CIGRE Networks[END_REF]. It is a Python-based tool combining the "pandas" data analysis library and the "pypower" power-flow solver to create a grid calculation program, aimed at automating analysis and optimization in electrical energy systems. The simulated grid used in the study is the CIGRE-Network medium voltage distribution with 14 nodes (Fig. 1). The publicly available consumption data "Smart meters in London" were used as hourly net consumption profiles of the simulated grid for two consecutive years (called year n-1 and year n) [START_REF]Smart meters in London[END_REF]. This database contains energy consumption for 5567 London households which took part in the "Low Carbon London" project led by British grid operators. The data covers period between November 2011 and February 2014. With a large number of real consumers over a large period of time, (and with virtually no lost measurement points), this dataset is perfectly suited for experiments. Data for the complete 2012 and 2013 years were extracted. Almost 900 consumption profiles were aggregated into 14 groups for each node of simulated grid (around fifty households per node). The data of solar radiation (typical meteorological year, TMY) for London from web application NREL's PVWatts [START_REF]PVWatts Calculator[END_REF] was used to model PV generation in some nodes. To get the temperature, the data DarkSky API [START_REF]Dark Sky API[END_REF] was used for the same geographic position and datetimes as the net consumption. PV generation profiles, modeled using solar radiation data, were added to seven random nodes in year n and scaled according to the objectives of the experiments. However, the solar radiation is used only for modeling and not as exogenous data for the detection. In this paper, the value of the PV installed capacity in the node is expressed as a ratio to the peak load value of this node / max( )

PV load nom P P

. There are 14 nodes for the simulation over two years, seven of which (arbitrary chosen) have PV added during the second year (Fig. 1). The goal is to develop an approach that can correctly detect the PV connection to these seven nodes of a distribution grid during a considered period (e.g. during the last year, month, or any other duration).

Method

The developed tool consists of a NN coupled with an analytical classification algorithm that operates separately for each node. Thus, it is not necessary to take into account the grid topology. However, Pandapower needs a topology to run a simulation and get the consumption-n profiles for each node. The operating principle of the method is shown in Fig. 2. The data in the first red rectangle (at the bottom) represents the year n-1 ("training set") and the data in the second (at the top) represents the year n, for which the recently installed PV shall be detected. Finally, the analytical classification method compares the baseline and measurement at the same hours throughout the day and then during only the sunshine in order to detect PV. After sensitivity studies, five features were selected for NN that show the strongest effect on consumption. The selected features were converted by hot encoding into 31 neurons of the first NN layer (Fig. 3). One hot encoding is a process by which categorical variables are converted into a form of bits, i.e., if a data point belongs to the category then corresponding component is 1, otherwise it is 0. In considered NN:

• Then Et S is similarly computed as the difference between the baseline and the measurement, but only for sunshine hours (between 9 a.m. and 4 p.m. of each day). Thereby the total considered period T S is equal to 2920 hours for the whole year or less if T < 8760 hours. The time period 9 am -4 pm is selected as period with noticeable solar radiation level whatever the season.

The algorithm then counts the number of hours N1 when Et exceeds the threshold value Emin (10% by default). In other words, it counts the number of hours for which the baseline is at least 10% greater than the measurement result.

If N1 is outside the interval of [10 %; 35 %] of the total number of hours in the considered period, the algorithm adjusts the threshold E min by 1 % to avoid an overestimation or underestimation from the NN and then repeats the process, counting N1 again until N1 falls into the interval.

Emin is the only threshold that needs adapting. The values of the interval were obtained empirically through sensitivity studies that experimentally determined the best percentages.

Once Emin is set, the algorithm calculates N2, the number of hours for which the baseline is greater than the measurement during the period T S by at least the value of the previously obtained threshold Emin.

Then the algorithm checks if N2/T S is at least L1(140%) times greater than N 1/T. Thus, it checks that the hours when the baseline is greater than the measurement on Emin are more frequent during the sunshine (period T S ) than during the whole period T. If there is no new installed PV, N2/T S and N1/T should be roughly the same. If there is new PV, then N2/T S will be significantly greater than N1/T.

Finally, the algorithm checks that N2/T S is greater than L2.

That is, at least during L2 (35 %) hours of T S baseline is greater than measurement by Emin%. This is necessary to avoid situations where, for instance, N1/T = 10 % and N2/T S = 15 %. Indeed, then, the previous condition is met, but there is no PV on the node, because the proportion of N2/T S is too small. If both of the last checks were successful, the algorithm concludes that there is a new PV installed in this node.

Results

As already mentioned, 14 nodes with different consumption profiles were prepared, with PV connected to seven of them, arbitrary selected (Fig. 1). The proposed tool was independently tested on each of these 14 nodes.

It is possible to increase the sensibility of the tool by analyzing only months with high level of solar radiation instead of the whole year. To confirm this hypothesis, two different cases are considered: application of the tool with consumption data over the six sunniest months (from April to September) and over the six least sunny months (from October to March).

The average accuracy was calculated for all possible periods of x days over 6 months with a rolling basis. (e.g. for "period = 40 days", the average accuracy of 142 possible 40-day periods between April and September was calculated).

The dependence of the average accuracy of the tool with respect to the period under consideration and to / max( )

PV load nom P P
for all nodes of the grid is presented in Fig. 5 (from April to September) and in Fig. 6 (from October to March). It can be concluded, that the average accuracy for the sunniest months is higher for longer periods, because the longer period allows to smooth out the impact of cloudy days, when PV generates less energy. It is also natural that a higher / max( )

PV load nom P P
makes it easier to detect PV, so the average accuracy is also higher.

The results are different from October to Mars. The average accuracy for these months is between 0.87 ( / max( ) < 8.5 % the average accuracy is higher for shorter periods, because on average during these months the PV systems do not generate enough power for detection, but there are few days with high solar radiation level when detection is possible. From Fig. 5 and Fig. 6 it can be concluded that, on average, three months are required to obtain the maximum detection accuracy. Considering the periods of the three sunniest months, 100 % accuracy can be achieved with values of The results of the metrics (for nodes with PV) that are used to detect PV are presented in Table 2. Data are given for the three sunniest months of the year (from May to July). The value of 1 2 / S N N T T for node 12 is equal 1.4, which means that further decreasing the installed PV capacity in that node will cause the tool to be unable to detect it. It should be mentioned that the sensitivity of the tool depends on consumption profiles (for two years), so for Node 11, for example, the tool can detect PV even with very small installed capacity ( / PV load nom W W < 3 %).

PV

Conclusion

The growing need for accurate prediction of consumption in distribution grid requires a way to detect locations with "hidden'' installed PV. The use of neural networks for capturing energy consumption behavior is very popular as the model can be trained offline and then quickly applied for any period of time.

The method proposed in the article for PV detection only needs data about temperature and net consumption that makes it more suitable for usage than other similar solutions, which require specific hard-to-find data (e.g., satellite images or solar radiation for specific spot). Moreover, the algorithm displays a high accuracy, which however depends on the period of consideration and the amount of installed capacity. Thus, it is necessary to find a compromise between the volume of available data and the required accuracy of the solution. It was found that the highest precision and accuracy could be reached for approximately a three months period under consideration. Future work may involve testing the tool on larger simulated grids. The next step will then be to not only detect PV installations in the nodes, but also to disaggregate the load and generation profiles, and hence approximate the installed PV capacity. Regarding the data, additional consideration could be given to noise and anomalies, because the simulated data has been very clean in the presented work.

Fig. 1 .

 1 Fig. 1. Medium voltage distribution network with PV (in red) installed in seven random nodes [7].

Fig. 2 .

 2 Fig. 2. Principle of operation of the method.Grid topology and real consumption data for n-1 year are first integrated into the Pandapower simulator. The simulator then generates hourly energy consumption per node. Besides, the corresponding temperature, time, and date data are added for each point of power consumption. Trained with those data for the year n-1, the NN produces the expected consumption for a given day and temperature of the year n at an hourly resolution, assuming that no new PV has been added. This forecast is called baseline, which is then fed to the classification algorithm. The net energy consumption measured by the meters (which measure the difference between load and PV generation) for the considered period of the year n is also supplied to the classification algorithm and is called measurement. Similar to the data for the NN training, those measurements were generated with the Pandapower tool, by taking into account the consumption of households and solar radiation. Finally, the analytical classification method compares the baseline and measurement at the same hours throughout the day and then during only the sunshine in order to detect PV. After sensitivity studies, five features were selected for NN that show the strongest effect on consumption. The selected features were converted by hot encoding into 31 neurons of the first NN layer (Fig.3).One hot encoding is a process by which categorical variables are converted into a form of bits, i.e., if a data point belongs to the category then corresponding component is 1, otherwise it is 0. In considered NN:• 24 neurons indicate the hour of the day for what the simulation will be performed, hi, where i = 1. ... 24; • Four others indicate the season, Sh (winter), Sp (spring), Se (summer), Sa (autumn); • The last three are temperature (T°), weekend (W) and holiday (F).

  24 neurons indicate the hour of the day for what the simulation will be performed, hi, where i = 1. ... 24; • Four others indicate the season, Sh (winter), Sp (spring), Se (summer), Sa (autumn); • The last three are temperature (T°), weekend (W) and holiday (F).

Fig. 3 .Fig. 4 .

 34 Fig. 3. The neural network and its features. The NN architecture with two hidden layers, of 20 and 15 neurons each, has shown the highest accuracy as the result of sensitivity studies. The output layer consists of only one neuron, which gives the expected consumption of the node for the chosen hour, given that no PV is connected. The NN is trained with the following parameters: Adam optimizer, sigmoid activation function for the all nodes, mean absolute error loss function, learning rate of 0.03 and 1000 epochs. The training time is 10-20 seconds per node.The analytical classification algorithm is presented in Fig.4. The basic idea is that, in absence of PV generation, an average difference between the baseline and the measurement values is roughly the same for hours of the day and of the night. Thereby, if this difference is greater during hours of the day, the algorithm detects the installation of new PV in the considered period, since the PV generation decreases the measurement values only during sunshine. First, the algorithm calculates Et (in %) as the difference between the baseline (Bt) and the measurement (Pt), for each hour t of period T.

  The average accuracy from April to September is between 00 %, for a period of more than two months).

Fig. 5 .

 5 Fig. 5. Average accuracy dependence from April to September.

Fig. 6 .

 6 Fig. 6. Average accuracy dependence from October to March.

Table 1 .Table 1 .

 11 6.2 % for the installed PV capacity in the node ( ) and 3.1 % for the generated PV energy compared to the energy consumption of the same node ( Distribution of the installed PV power and the corresponding energy production.

	PV nom P	/ max(	P	load	)
	/ W ), as shown in Bus PV load nom W 2 3	4	8	9	11	12
	PV nom P max( load P	)	,%	6,2 6,2 6,2 6,2	6,2	6,2	6,2
	PV nom load W W	,%					3,25 3,2 3,3 3,1 3,05 3,05 3,68

Table 2 .

 2 Results for nodes with PV from May to July 2013.

	Bus	2	3	4	8	9	11	12
	1 N	34,4 33,6 34,6 33,5 34 33,5 32,8
	T
	N	2	50,6 59,3 60,1 57 62,3 54	46
	T	S
	1 T T / S 2 N N	1,47 1,76 1,74 1,70 1,83 1,61 1,40