Fellow, IEEE Bernard Picinbono
email: bernard.picinbono@lss.supelec.fr

Patrick Duvaut

Geometrical Properties of Optimal Volterra Filters for Signal Detection

Volterra filters are a particular class of nonlinear filters defined by an extension of the concept of impulse response to the nonlinear case. Linear-quadratic filters are a special example of Volterra filters limited to the second order. In the first part of this paper it is shown that all the results recently published, valid in the linear-quadratic case, can be extended with the appropriate notations to Volterra filters of arbitrary order. In particular, the optimum Volterra filter giving the maximum of the deflection for the detection of a signal in noise is wholly calculated. In the second part several geometrical properties of optimal Volterra filters are investigated by introducing appropriate scalar products. In particular the concept of space orthogonal to the signal and the noise alone reference (NAR) property are introduced allowing a decomposition of the optimal filter that exhibits a relation between detection and estimation.

Extensions to the infinite case and relations with the likelihood ratio are also investigated.

I. INTRODUCTION

THE CONCEPT of an optimal linear-quadratic system for detection and estimation was introduced in a previous paper [l]. It was also noted that linear-quadratic systems can be described by a Volterra expansion limited to the second order term. Volterra filters (VF) have been introduced into many aspects of nonlinear filtering problems, and we shall not mention here all the papers discussing their properties.

For an introduction to this field applied to problems similar to those discussed hereafter, the reader should consult [21-[51. It is thus natural to extend the results presented in [l] to VF of arbitrary order, and this is the first purpose of this paper. At first glance this extension is quite obvious [START_REF] Duvaut | Détection, estimation et filtres de Volterra[END_REF], and to avoid excessive length, we will present the main ideas and fundamental results. Our principal effort will be to introduce simplified notations in such a way that the analogy between the second order case and the general case appears evident. In fact, the input-output relationship of a VF is extremely tedious to write out, and without our simplified notation the paper would be full of complex equations contributing little to the understanding of the physical meaning of the results. It should be clear, however, that the exact analytical solution of our condensed equations would require extensive calculations.

The extension of a second order to an arbitrary order VF exhibits new problems which were not analyzed in the linear-quadratic case. Among these problems we are mainly interested in the relation between optimal VF and the likelihood ratio (LR) filter that is the absolute optimal system for the test, or the detection, between two simple hypotheses, say the null hypothesis H 0 corresponding to the noise only situation, and the hypothesis H 1 corresponding to the presence of a signal. For this purpose filters are considered as vectors in appropriate Hilbert spaces. In particular it is shown that the filters belonging to the space orthogonal to the LR have the NAR property. This is the basis of a decomposition of the optimal VF in two terms that exhibits relations between detection and estimation. Some extensions to the infinite case are also discussed.

II. OPTIMAL VOLTERRA FILTERS

A. Principles of Volterra Filtering

A Volterra filter of order v (v positive integer) is a nonlinear system the input-output relationship of which is described by a finite Volterra expansion. To simplify the presentation we will consider the discrete time case only, and we call x[k] and y[k] the input and output, respectively. The output is expressed by

y[k] = h 0 + v m=1 {i m n } h m [i m 1 , i m 2 , . . . , i m m ]x[k -i m 1 ]x[k -i m 2 ] . . . x[k -i m m ]. (1)
In this expression k and i m n are arbitrary integers and the symbol {i m n } means that the sum is taken on all the i m n s. The term h 0 corresponding to m = 0 is a constant and is sometimes omitted. In this latter case, and if v = 1, we obtain the classical convolution characterizing linear filters. By an obvious extension of the terminology used in the linear case, we will say that the Volterra filter has a finite impulse response if the integers i m n are taken in a finite interval, as, for example, 1 ≤ i m n ≤ N. We make this assumption henceforth to avoid convergence problems. In this case the input at time k can be considered as a vector x with components x[i], 1 ≤ i ≤, and the output can be written as

y[k] = h 0 + v m=1 {i m n } h m [i m 1 , i m 2 , . . . , i m m ]x[i m 1 ]x[i m 2 ] . . . x[i m m ] (2)
where the i m n s satisfy 1 ≤ i m n ≤ N . For the discussion that follows it is necessary to simplify the notations a little more, and to this end we first write

h m [i m 1 , i m 2 , . . . , i m m ] = h m [i m ]. (3)
Furthermore we associate with the v kernels h m the quantity

|h, v >= {h 1 [i 1 ], h 2 [i 2 ], . . . , h v [i v ]}. (4)
The quantity λ|h, v > is obtained by replacing the hs by λh and the sum |h, v > +|h , v > is obtained by replacing the hs by h + h . With these rules the quantity|h, v > becomes a vector of a linear space called R N [v]. For v = 1 and without a constant term we find the classical vector space R N .

As in any linear vector space, we can introduce linear operators such as

A|h, v >= |h, v > (5)
and all the concepts associated with these operators are valid in

R N [v].
More important is the concept of a scalar product of vectors of R N [v]. This scalar product of two vectors |h, v > and |h, v > is defined by

< h, v|h , v >= v m=1 im h[i m ]h [i m ] (6)
which is a direct extension of that used in R N [v]. This allows the introduction of the transpose of an operator A defined by

< g, v|A|h, v >=< h, v|A T |g, v > . (7)
Finally, an operator A is said to be symmetric if A = A T and positive definite if for any |h, v > we have

< h, v|A|h, v >≥ 0. (8)
July 4, 2018 DRAFT

Let us now define the input vector |X, v >. The input signal in ( 2) is a real N-dimensional vector x with components x[i] . To this vector of R N we associate the vector of R N [v] defined by

|X, v > = {x[i 1 1 ], x[i 2 1 ]x[i 2 2 ], . . . , x[i v 1 ]x[i v 2 ] . . . x[i v v ]} (9)
It is the v order input vector associated with the input x. With this notation (2) can be written as

y = h 0 + < h, v|X, v >, (10)
which greatly simplifies the notations. Of course the exact calculation of y requires the return to [START_REF] Alper | A consideration of discrete Volterra series[END_REF]. It

is clear that if v = 1 or v = 2
we again find the linear and linear-quadratic filters. Finally, to simplify a little more, we will omit the letter v when no ambiguity is possible in the order of the filter. Thus in the following |h > means |h, v >, whenever the order v of the filter is well specified.

In detection and estimation problems the input x is usually random. When this is the case, the vector |X > also becomes random, as does the output y given by [START_REF] Blanc-Lapierre | Fonctions Aléatoires[END_REF]. The randomness of the input introduces two points that must be briefly discussed. First, and most important, the output y can be undefined, in the sense that it is not a second order random variable. To avoid this situation we must assume that the input vector x has finite moments up to order 2v. This ensures that y has a finite second order moment, or is a second order random variable. Secondly, y does not in general have a zero-mean value. To define this value let us introduce

|m > = E(|X >) (11)
which is a vector of R N [v] obtained by taking the expectation value of the terms of [START_REF] Kailath | Lectures on Linear Least-Squares Estimation[END_REF]. Then

E(y) = h 0 + < h|m >, (12)
and if h 0 = -< h|m >, y becomes, of course, zero-mean. For any arbitrary |h > it is then possible to use the constant term h 0 in the Volterra expansion (2) in such a way that y becomes zero-mean. This procedure is equivalent to using the vector |X 0 > defined by

|X 0 >= |X > -|m >, (13)
which is, of course, zero-mean and to writing [START_REF] Blanc-Lapierre | Fonctions Aléatoires[END_REF]

B. Optimal Volterra Filter for Detection

As in [l], our purpose is to find the optimal filter giving the maximum of the deflection of the output y defined by

D(y) = N/D, (14)
N = [E 1 (y) -E 0 (y)] 2 , (15)
D = V 0 (y), (16)
where E 0 and E 1 mean-expectation value under hypotheses H 0 and H 1 respectively and V 0 denotes the variance under H 0 .

To calculate the numerator we introduce the notation

E 1-0 (y) = E 1 (y) -E 0 (y), (17)
and we define the signal vector by

|s > = E 1-0 (|X >) = E 1-0 (|X 0 >). (18)
With this notation we get

N =< h|s > 2 . ( 19
)
As y is zero-mean under H 0 , its variance is E 0 (y 2 ), or

V 0 (y) = E 0 [< h|X 0 >< X 0 |h >], (20)
where |X 0 > is defined by [START_REF] Astrom | Introduction to Stochastic Control Theory[END_REF]. Note that as the expectation values in [START_REF] Sakrison | Communication Theory; Transmission of Waveforms and Digital Information[END_REF], [START_REF] Astrom | Introduction to Stochastic Control Theory[END_REF], and (20) are taken under H 0 y is zero-mean under H 0 but not under H 1 , which appears in [START_REF] Journel | Kriging in terms of projections[END_REF] and [START_REF] Picinbono | A geometrical interpretation of signal detection and estimation[END_REF]. Finally the variance can be written as

V 0 (y) =< h|K|h >, (21)
with

K = E 0 [|X 0 >< X 0 |]. (22)
This is a positive definite symmetric operator. With these notations and by using the Schwarz inequality we find that the maximum value of the deflection obtained with a Volterra filter of order v is

d 2 v =< s, v|K -1 |s, v >, (23)
and the corresponding optimal filter is

T o,v (x) = α < s, v|K -1 |X 0 , v > . ( 24
)
We will assume henceforth that α = 1. This result gives the classical matched filter [71 for Lv = 1, and the optimal linear-quadratic filter for v = 2. The exact calculation of this filter requires the solution of the equation

K|s >= |s >, (25)
which can, of course, be a tedious job for high values of v. As a result, T o,v (x) takes the form The quantity < a|M |b > satisfies all the requirements necessary to define a scalar product, written as

T o,v (x) =< s|X 0 > . (26)
< a|b > M =< a|M |b > . (27)
The space R N [v, M ] is the space deduced from R N [v] in which the scalar product of vectors is defined by ( 27). We obviously have

R N [v, I] = R N [v],
where I is the identity operator. In this space, orthogonal vectors are called orthogonal (⊥, M ) and denoted by

< a|b > M = 0 ⇔< a|M |b >= 0 ⇔ |a >⊥ (M )|b > . ( 28
)
As the projection is a concept deduced from orthogonality, we will use the term projection (⊥, M ).

In other words the projection (⊥, M ) onto a subspace S is written proj(⊥, M)[|a > |M] and defined by the orthogonality principle, or

{|a > -proj(⊥, M )[|a > |S]} ⊥ (M )|b > (29)
for any vector |b > of S.

B. The Space H v

The output y(x) of a VF considered in Section I1 is a zero-mean second-order random variable written as y(x) =< h, v|X 0 , v >. This output can then be considered as a vector of the space L 2 of scalar second order random variables. In this space the scalar product between two vectors u and v noted

(u, v) = E 0 [uv],
and the orthogonality as well as the projection is written with the symbol ⊥ ⊥, to avoid any confusion with the concepts introduced before. Note that the expectation value is taken under the hypothesis H 0 as defined after [START_REF] Stirling | Least squares subject to linear constraints[END_REF]. The space H v . is the subspace of L 2 containing all the outputs of VF with zero-mean value, or

H v = {F (x)|F (x) =< h, v|X 0 , v >}, F (x) ∈ L 2 }. ( 30
)
DRAFT July 4, 2018

C. The Signal Subspace and Its Complement

Consider the vector |s > of the space R N [v, M ] defined by [START_REF] Ayadi | NAR AGC adaptive detection of nonoverlapping signals in noise with fluctuating power[END_REF] and the subspace R N ⊥ [v, M ; s] of vectors of R N [v, M ] that are orthogonal (⊥, M ) to |s >. By using the projection theorem, any vector

|a > of R N [v, M ] can be written as |a >= a s |s > +|a, s ⊥ >, (31)
where

|a, s ⊥ >= proj(⊥, M )|a > | R N ⊥ [v, M ; s]. (32)
Furthermore, the component a s given by

a s =< s|M |s > -1 < s|M |a > . (33)
It is obvious that

|s >⊥ (M )|a, s ⊥ > . (34)

D. The Signal Decomposition of the Optimal Filter

The optimal filter (26) can also be written as

T o (x) =< σ|M |X 0 >=< σ|X 0 > M , (35)
where

|σ >= M -1 K -1 |s >= M -1 |s > . (36)
Applying the decomposition (31) to the vectors |σ > and |X 0 > of (35) and taking into account the various orthogonality (⊥, M ) properties, we get

T o (x) = T s (x) + T s,⊥ (x), (37)
with

T o (x) = < s|M |σ > < s|M |s > < s|M |X 0 >= < s|K -1 |s > < s|M |s > < s|M |X 0 >= d 2 v < s|M |X 0 > < s|M |s > . (38)
The term T s,⊥ (x) is deduced from (37) with the values (35) and (38) for T o (x) and T s (x), respectively.

Let us investigate a little more precisely, in a particular case, the meaning of the term T s (x). Suppose that M = I , which is the more common situation. We deduce from (38)

T s (x) = d 2 v s -2 < s|X 0 >, (39)
July 4, 2018 DRAFT where d 2 v is given by (23) and s 2 =< s|s >. Suppose now, for simplification, that v = 2, which corresponds to linear-quadratic systems studied in [ l l . In this case, we obtain easily

T s (x) = d 2 v s -2 [s T x + x T Kx -tr(CK)], (40)
where

s = E 1-0 [x] and C = E 0 [xx T ].
The first term in the bracket (40) is the output of the matched filter in a white noise, sometimes also called correlation receiver, because it performs a correlation between the input observation and the given vector s. The second term plays a similar role, but for a quadratic receiver. In conclusion, the term T s (x) uses directly the primary data on the hypotheses (s, K, C) without any particular processing. This processing is performed through the term T s,⊥ (x).

IV. GEOMETRY IN L 2 FOR THE OPTIMAL VOLTERRA FILTER

A. Basic Relations and the Deflection Property

Let us assume that the observation vector x(ω) of R N is a random variable with probability densities p 0 (x) and p 1 (x) under H 0 and H 1 , respectively. The LR appearing in the statistical decision theory is defined by

L(x) = p 1 (x)/p 0 (x). (41)
We assume in what follows that L((x)) nL 2 which means that

(L, L) = E 0 (L 2 ) = l 2 < ∞. (42)
There are some probability densities for which this is not satisfied. Such situations require a different approach from the following and are thus outside the scope of our discussion.

In detection problems it is interesting to introduce the displaced LR, R(x), defined by [8]

R(x) = L(x) -1. (43) It is clear that E 0 (R) = 0 and (R, R) = l 2 -1 = d 2 . ( 44
)
Property 4.1 : Any function F (x) of L 2 satisfies the "deflection property" characterized by

(R, F ) = E 1-0 [F (x)]. (45)
Proof : It is a direct consequence of the definition of the scalar product by E 0 R(xR(x) where R(x is replaced by (43).

Comments : The term "deflection" is due to the fact that (45) gives the numerator of the deflection criterion, as seen in [START_REF] Journel | Kriging in terms of projections[END_REF].

DRAFT July 4, 2018

As all the components of |X, v > defined by ( 9) are vectors of L 2 , we deduce immediately from [START_REF] Ayadi | NAR AGC adaptive detection of nonoverlapping signals in noise with fluctuating power[END_REF] that |s > may be written as

|s >= (R, |X >) = (R, |X 0 >). (46)
We can apply that to (35) and (38), which gives

E 1-0 [T o (x)] = E 1-0 [T s (x)] = d 2 v , (47)
where d 2 v is defined by (23).

B. The NAR Subspace

Let us introduce the subspace of L 2 defined by

H ⊥ ⊥ = {F (x)|(F, R) = 0}, (48)
which is then the subspace of functions of the observation x orthogonal (⊥ ⊥) to R. It is clear from (45) that any vector of this subspace satisfies

E 1 (F ) = E 0 (F ), (49)
which is called the NAR (noise alone reference) property [71, [START_REF] Kailath | Lectures on Linear Least-Squares Estimation[END_REF]. This means that the mean value of F is the same under H 0 and H 1 , and as H 0 is the noise only hypothesis, while H 1 corresponds to the signal-plus-noise, the mean value of F has a property of the noise alone. Furthermore, it is clear from [START_REF] Journel | Kriging in terms of projections[END_REF] that any function of this space gives a null deflection.

It is important to note that the NAR property is only valid in the mean, as seen on (49). It is sometimes possible to introduce an almost surely NAR property, which is especially the case for v = 1 and for the detection of a deterministic signal in noise. In fact the observation vector is n(ω) under

where n s (ω) is the projection of the noise vector n s (ω) onto the subspace orthogonal to s. From this it is clear that n s,⊥ (ω) is independent of the presence or absence of a signal.

Property 4.2 : The function T s,⊥ (x) introduced in (37) belongs to the NAR subspace.

Proof : From (37), we get T s,⊥ (x) = T o (x)-T s (x). As a result of (47), we deduce that E 1-0 [T s,⊥ (x)] = 0 and this gives with (45) (R, T s,⊥ (x)) = 0, which characterizes a vector of the NAR subspace.

July 4, 2018 DRAFT

  with h 0 = 0. This is done in what follows. This was already the case in [l] where linear-quadratic systems were written as y = h T x + x T Mx -Tr(CM), where x has zero-mean value and covariance matrix C. DRAFT July 4, 2018

  III. GEOMETRICAL CONCEPTS FOR FINITE-ORDER VOLTERRA FILTERSA. The Space R N [v, M ] Consider an arbitrary symmetric positive definite operator M and two vectors |a > and |b > of R N [v].

H 0 and n(ω) + s under H 1 .

 1 If we use (31) in R N = R(1, I), we obtain n(ω) = n s (ω) + n s,⊥ (ω),

July 4, 2018 DRAFT

C. Optimal VF and the Likelihood Ratio Property 4.3 : The projection ⊥ ⊥ of the function R(x) given by (43) onto the subspace H v defined by (35) is the optimal filter (401, or

Proof : We have to show that for any element G v (x) of H v I defined by

we have

This last equation can be written

By using (46), this equation becomes

which gives the result.

Comments : This property establishes the relation between the likelihood ratio L(x), or its displaced version R(x), and the optimal Volterra filtering. It is a consequence of the fact that the filter with zeromean value that maximizes the deflection without any constraint is R(x), [START_REF] Papoulis | Signal Analysis[END_REF]. As the deflection can be interpreted as a distance [START_REF] Papoulis | Signal Analysis[END_REF], it is natural that the optimal Volterra filter is the projection of R(x), onto the Volterra subspace.

Furthermore note that in the linear case

In fact the orthogonality relation (54) easily yields

where Γ = E 0 (xx T ). This is the equation defining the classical linear matched filter. Of course, as

and the matched filter is the projection of the LR, or its displaced version, onto the subspace of linear filters. Various other estimation problems can also be solved by this geometrical description of Volterra filtering and will be analyzed in other publications.

DRAFT July 4, 2018


Marina Bertolini

AND Roberto Notari
email: roberto.notari@polimi.it

Christina Turrini

Cristina Turrini
email: cristina.turrini@unimi.it

THE BORDIGA SURFACE AS CRITICAL LOCUS FOR 3-VIEW RECONSTRUCTIONS

Keywords: 2010 Mathematics Subject Classification. 14J25, 14M12, 14M15, 14N05 Bordiga surface, Line congruences in Grassmannians, Projective reconstruction in Computer Vision, Multiview Geometry, Critical configurations or loci

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Linear projections from P 3 to P 2 are the natural geometric model for pictures of static three-dimensional scenes taken from pinhole cameras. Similarly linear projections from P k to P 2 , with k ≥ 3, arise when images of particular dynamic and segmented scenes are considered ( [START_REF] Wolf | On projection matrices P k → P 2 , k = 3, ..., 6 and their applications in computer vision[END_REF][START_REF] Hartley | Reconstruction from projections using Grassmann tensors[END_REF][START_REF] Kun Huang | Generalized rank conditions in multiple view geometry with applications to dynamical scenes[END_REF][START_REF] Fan | The space of multibody fundamental matrices: Rank, geometry and projection[END_REF][START_REF] Hartley | The multibody trifocal tensor: Motion segmentation from 3 perspective views[END_REF]).

Given multiple images of an unknown scene, taken from unknown cameras, the reconstruction of the positions of cameras and scene points is a classical problem in Computer Vision, which has been generalized as well in the setting of higher dimensional projective spaces.

Sufficiently many images and sufficiently many sets of corresponding points in the given images should in principle allow for a successful projective reconstruction. Anyway, there exist sets of points, in the ambient space P k , for which the projective reconstruction fails. These configurations of points are called critical, which means that there exist other non projectively equivalent sets of points and cameras that give the same images in the view planes.

Critical loci for projections from P 3 to P 2 have been studied by many authors. Among the many papers on the subject, we recall [START_REF] Buchanan | The twisted cubic and camera calibration[END_REF][START_REF] Krames | Zur ermittlung eines objectes aus zwei perspectiven (ein beitrag zur theorie der "gefhählichen Örter[END_REF][START_REF] Maybank | Theory of Reconstruction from Image Motion[END_REF][START_REF] Hartley | Ambiguous configurations for 3-view projective reconstruction[END_REF][START_REF] Kahl | Critical configurations for n-view projective reconstruction[END_REF][START_REF] Hartley | Critical configurations for projective reconstruction from multiple views[END_REF][START_REF] Shashua | Degenerate n point configurations of three views: Do critical surfaces exist? TR 96-19[END_REF][START_REF] Åström | Ambiguous Configurations for the 1D Structure and Motion Problem[END_REF]. In the case of projections from higher dimensional P k to the projective plane P 2 , when k ≥ 4, critical loci were described in [START_REF] Bertolini | Critical configurations for 1-view in projections from P k → P 2[END_REF] in the case of one view, and in [START_REF] Bertolini | Critical loci for projective reconstruction from multiple views in higher dimension: A comprehensive theoretical approach[END_REF][START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] in the case of multiple views.

In this paper we focus on the case of three projections from P 4 to P 2 , that is the first nonclassical case, since three views is the minimum number which allows us to reconstruct the scene, when the scene points are general (in a sense which will be clear later). Our purpose is to get a scheme-theoretical description of the critical locus. The critical locus comes out to be a classical surface in P 4 , the so-called Bordiga surface ( [START_REF] Bordiga | La superficie del 6 • ordine con 10 rette, nello spazio R 4 e le sue proiezioni nello spazio ordinario[END_REF]). The approach used here to obtain the polynomials that generate the ideal of the critical locus is different from the one followed in [START_REF] Bertolini | Critical loci for projective reconstruction from multiple views in higher dimension: A comprehensive theoretical approach[END_REF]: we use the Grassmann tensor introduced in [START_REF] Hartley | Reconstruction from projections using Grassmann tensors[END_REF]. In [START_REF] Åström | Ambiguous Configurations for the 1D Structure and Motion Problem[END_REF], a first seminal case of this idea has been applied to two projections from P 2 to P 1 , while in [START_REF] Bertolini | Critical hypersurfaces and instability for reconstruction of scenes in high dimensional projective spaces[END_REF] this approach is used to study the critical locus when it is a hypersurface. The construction of the critical locus given in [START_REF] Bertolini | Critical loci for projective reconstruction from multiple views in higher dimension: A comprehensive theoretical approach[END_REF] allowed the set-theoretical description of it, while the one given here through the Grassmann tensor allows us to compute the generators of the ideal of the critical locus and its first syzygy module, so giving a scheme-theoretical description of the critical locus itself. In more details, the ideal is minimally generated by 4 degree 3 forms that are the maximal minors of a 4 × 3 matrix with linear entries. From the Hilbert-Burch Theorem, it follows that the critical locus is a determinantal variety of codimension 2 and degree 6 in P 4 , and so it belongs to the irreducible component of the Hilbert scheme containing Bordiga surfaces.

A very natural question arising as a consequence of the above results is whether every Bordiga surface in P 4 is the critical locus of suitable projections. To give a positive answer to this question, we heavily use the geometry of Bordiga surfaces.

We recall that a Bordiga surface S is the blow-up of P 2 at 10 general points, embedded in P 4 via the complete linear system of the quartics through the 10 points. S contains exactly 10 lines, corresponding to the base points of the linear system. The ideal of the 10 points is determinantal, too. The 5 × 4 matrix of linear forms whose maximal minors are the generators of the ideal of the 10 points in P 2 is strongly related to the matrix that presents the surface S in P 4 . We use such a relation to prove that the generators of the first syzygy module of the ideal of S can be chosen in a very particular form. Finally, we use this particular presentation to prove that every Bordiga surface is the critical locus for suitable projections, that we explicitly construct.

Quite surprisingly, the Bordiga surface is linked to another classical problem in Computer Vision: reconstruction using lines. Given an unknown scene in P 3 , consisting of a set of lines (i.e. a subset of the Grassmannian G(1, 3)), and taken multiple images of this scene, in which corresponding lines in the different views are identified, the goal is the reconstruction of the scene. The reconstruction using lines is particularly significant in the real word, since images of lines can be more easily and accurately detected and tracked than points [START_REF] Maybank | The critical line congruence for reconstruction from three images[END_REF]. A classical result of Buchanan and Maybank shows that the critical locus for reconstruction in P 3 using lines is essentially a line congruence of bidegree [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] and sectional genus 5 in G [START_REF] Arrondo | Sols On conguences of lines in projective space[END_REF][START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF]. This congruence admits a biregular model in P 4 which is a Bordiga surface [START_REF] Verra | Smooth surfaces of degree 9 in G(1, 3)[END_REF]. Furthermore, Maybank proves that every Bordiga surface can be obtained in this way, i.e. as model in P 4 of a congruence in G(1, 3) which arises as the critical locus for the reconstruction using lines. As a consequence of this, we have more deeply investigated, also from an algebraic point of view, the relations between the two critical loci. To do this we have computed the generators of the ideal of the critical congruence, obtaining its scheme-theoretical description, and proved that the whole critical locus in G(1, 3) is the union of the quoted congruence with three α-planes corresponding to the lines through the centers of projection.

Finally we have explicitly described how to link the projections from P 3 to P 2 used in the reconstruction using lines, with the ones from P 4 to P 2 used in the reconstruction using points, which give rise to biregular critical loci, and conversely. This provides a connection between the two reconstructions, a priori independent each other.

The plan of the paper is as follows. In sections 2 and 3 we recall the geometric constructions of the Bordiga surface and of the [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] line congruence respectively and, for both of them, we give the algebraic translation of their constructions. Moreover in section 3 we present some results on the [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF]) line congruence we are not able to quote. Section 4 is devoted to recall the main facts on multiview geometry in higher dimension, in particular the description of the Grassmann tensor. In section 5 we prove that the generators of the ideal of the critical locus for 3 projections from P 4 to P 2 can be deduced via the appropriate Grassmann tensor and that this locus is actually a Bordiga surface. Moreover we prove also that the converse holds, i.e. every Bordiga surface S is the critical locus of suitable linear projections. In section 6 we provide the definition of critical locus for reconstruction using lines, we compute the defining ideal of such a critical locus and finally we show that the [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] congruence is the main component of this locus. Section 7 gives a bridge between the two above problems of criticality, by explicitly computing the equations of the maps between the two critical loci. From the maps we are able to compute the projections of one of the two settings given the projections of the other one. In section 8 we provide a Singular session in which we compute a degree 10 arithmetically Gorenstein curve contained in a given [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] line congruence. We need this curve to make effective the algorithm in section 7.

On the Bordiga surface

In this section, we recall the construction of the Bordiga surface and some properties of this surface we'll use later on in the paper. For more information on this surface, see [START_REF] Bordiga | La superficie del 6 • ordine con 10 rette, nello spazio R 4 e le sue proiezioni nello spazio ordinario[END_REF], [START_REF] Dolgachev | Classical Algebraic Geometry. A modern view[END_REF].

Let

P 2 = Proj(S = C[z 0 , z 1 , z 2 ]
) be the projective space of dimension 2 over the complex ground field C, and let p 1 , . . . , p 10 ∈ P 2 be ten general points. Let P 2 π -→ P 2 be the blowup of P 2 in p 1 , . . . , p 10 . The linear system

|4π * L -E 1 -• • • -E 10 |
, where L is the line divisor in P 2 and E i is the exceptional divisor associated to p i , i = 1, . . . , 10, embeds P 2 in P 4 = Proj(R = C[y 0 , . . . , y 4 ]), and becomes the hyperplane divisor H of the image. If ϕ : P 2 → P 4 is the embedding, B = ϕ( P 2 ) is named Bordiga surface. B is a smooth surface of degree 6 and sectional genus 3. ϕ maps each E i onto a line, and moreover B contains exactly these 10 lines. It is possible to prove that these lines are pairwise skew and that no three of them are contained in a 3-dimensional linear space. The canonical divisor of

B is K B = | -3π * L + E 1 + • • • + E 10 |.
By adjunction, it is possible to prove that the image of a plane cubic through 9 among p 1 , . . . , p 10 is a plane cubic in B, as well. Moreover, the plane containing it and the exceptional line associated to the 10th point are skew linear spaces. Finally, the image of a line containing no point among p 1 , . . . , p 10 is a rational normal quartic curve Γ contained in B. Now, we give the algebraic translation of the previous geometric constructions.

Because of the generality assumption, Z = {p 1 , . . . , p 10 } has Hilbert function h Z (j) = min{ j+2 2 , 10} for j ≥ 0, and so I Z , defining ideal of Z in S, is generated by 5 forms g 0 , . . . , g 4 of degree 4. As Z is a codimension 2 arithmetically Cohen-Macaulay (ACM, for short) closed subscheme of P 2 , the ideal I Z is determinantal and its minimal free resolution is [START_REF] Arrondo | Sols On conguences of lines in projective space[END_REF] 0 → S 4 (-5)

N Z -→ S 5 (-4) → I Z → 0.
The forms g 0 , . . . , g 4 are a basis of the complete linear system of quartic curves through p 1 , . . . , p 10 . If we define ϕ : R → S as ϕ(y i ) = g i , i = 0, . . . , 4, then I B = ker(ϕ) is the defining ideal of the Bordiga surface B = ϕ( P 2 ). I B is determinantal too, and its minimal free resolution is

(2) 0 → R 3 (-4) N B -→ R 4 (-3) → I B → 0.
This proves that B is a codimension 2, ACM closed scheme in P 4 with Hilbert polynomial p B (t) = (6t 2 + 2t + 2)/2. From [START_REF] Ellingsrud | Sur le schéma de Hilbert des variétés de codimension 2 dans P e à cône de Cohen-Macaulay[END_REF], it follows that Proposition 2.1. The Bordiga surface is the general element of the irreducible component of the Hilbert scheme Hilb p B (t) (P 4 ) containing the codimension 2, ACM closed subschemes of P 4 with Hilbert polynomial p B (t).

From the point of view of liaison theory, the Bordiga surface is linked to a degree 3 rational normal scroll. In fact, let S be the surface complete intersection two general cubic hypersurfaces containing B. Then, by standard argument from liaison theory, the defining ideal of the residual surface B has the following minimal free resolution

0 → R 2 (-3) → R 3 (-2) → I B → 0
and so B is a degree 3 rational normal scroll, as claimed. Moreover, the intersection between B and B is an arithmetically Gorenstein curve of degree 8 and socle degree 3, as its minimal free resolution is

0 → R(-6) → R 2 (-3) ⊕ R 3 (-4) → R 3 (-2) ⊕ R 2 (-3) → I B∩B → 0,
computed by mapping cone from the short exact sequence

0 → I S → I B ⊕ I B → I B∩B → 0.
Such a curve is the image in B of a curve in the linear system

|7π * L -2E 1 -• • • -2E 10 | = |-K B +H|
and so the curve is a twisted anticanonical divisor on B (see, [START_REF] Kleppe | Gorenstein Liaison, Complete Intersection Liaison Invariants and Onubstructedness[END_REF] for generalities on such divisors). Such a linear system will play a role in next sections.

By adjunction ( [START_REF] Ottaviani | Varietà proiettive di codimensione piccola[END_REF], pp.48-49), one can prove that the matrices N Z and N B are related each other by If p ∈ Z, then rank(N Z ⊗ C(p)) = 3, and so the solutions of (4) are the points of the exceptional line E P . Moreover, if p ∈ Z, then I Z : I p has a unique generator in degree 3, the defining form f C of the plane cubic curve C through Z \ {p}. For every line L p, we have that lf C ∈ g 0 , . . . , g 4 , with l ∈ C[z 0 , z 1 , z 2 ] 1 defining L, and so lf c = a 0 (l)g 0 + • • • + a 4 (l)g 4 for suitable a i (l) ∈ C. Then, the hyperplane a 0 (l)y 0 + . . . a 4 (l)y 4 = 0 contains ϕ(C). As there are two linearly independent lines in P 2 containing p, we have that ϕ(C) is contained in a dimension 2 linear space, as well. Hence, ϕ(C) is a plane cubic, because I B is generated in degree 3. Now we want to explain why the above plane containing ϕ(C) is skew with E p . In fact, the solutions of (4) define the sub-linear system of plane quartics through p 1 , . . . , p 10 , singular at p. As the quartics defined by lf C are smooth at p, then they are not in the sub-linear system above, and so the intersection of the two linear spaces is empty. Now, if L ⊂ P 2 is a general line, we can assume that L ∩ Z = ∅. Let p, q ∈ L be two distinct points and let

(3) N B   z 0 z 1 z 2   = N T Z   y 0 . . .
M L =   z 0p z 0q z 1p z 1q z 2p z 2q
  be the rank 2 matrix whose columns are homogeneous coordinates of the points p, q. Then N B M L is a 4 × 2 matrix of generic rank 2 of linear forms in R, and so its maximal minors define a rational normal curve Γ of degree 4 contained in B.

3. On the line congruence of bi-degree [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] and sectional genus 5

In this section, we recall the construction and some properties of the line congruence K of bi-degree [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] and sectional genus 5 in the Grassmannian G(1, 3) of lines in P 3 , following [START_REF] Verra | Smooth surfaces of degree 9 in G(1, 3)[END_REF], [START_REF] Arrondo | Sols On conguences of lines in projective space[END_REF]. Furthermore, we present three properties of K for which we are not able to quote a reference.

As in section 2, let P 2 be the blow-up of P 2 at 10 general points p 1 , . . . , p 10 . The linear system |7π * L -2E 1 -• • • -2E 10 | on P 2 embeds P 2 in G(1, 3) ⊂ P 5 = Proj(T = C[x 0 , . . . , x 5 ]). For a suitable choice of the coordinates, G(1, 3) = V (x 0 x 5 -x 1 x 4 + x 2 x 3 ). Let ψ : P 2 → G(1, 3) be such an embedding, and let K = ψ( P 2 ) be its image. Then, K is a line congruence of bi-degree [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] and sectional genus 5. The canonical divisor is

K K = |-3π * L+E 1 +• • •+E 10 | while the hyperplane divisor is H = |7π * L-2E 1 -• • •-2E 10 |.
Moreover, the converse holds true, that is to say, every such a surface is the embedding of

P 2 with |7π * L -2E 1 -• • • -2E 10 |.
In G(1, 3), the ideal sheaf I K|G(1,3) is generated in degree 3 and it holds [START_REF] Bertolini | Reconstruction of some segmented and dynamic scenes: trifocal tensor in P 4 , theoretical set up for critical loci and instability[END_REF] 0

→ O 5 G(1,3) → E 2 (1) 3 → I K|G(1,3) (3
) → 0 where E 2 is the rank 2 vector bundle on G(1, 3) coming from the universal exact sequence

0 → E 1 → H 0 (O P 3 (1)) ⊗ O G(1,3) → E 2 (1) → 0.
The first result we are not able to quote concerns the shape of the minimal free resolution of the defining ideal I K of K in T .

Proposition 3.1. The minimal free resolution of I K has the shape

(6) 0 → T 5 (-5) → T 12 (-4) → T (-2) ⊕ T 7 (-3) → I K → 0.
Proof. From (5), we get the resolution of I K|G [START_REF] Arrondo | Sols On conguences of lines in projective space[END_REF][START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF] over T G = T / x 0 x 5 -x 1 x 4 + x 2 x 3 , that is periodic, and it is

• • • → T 12 G (-5) → T 12 G (-4) → T 7 G (-3) → I K|G(1,3) → 0 as H 0 (E 2 (1)) = H 0 (O P 3 (1))
. By [START_REF] Shamash | The Poincaré series of a local ring[END_REF], the minimal free resolution of I K has the shape

0 → T 5 (-5) → T 12 (-4) → T (-2) ⊕ T 7 (-3) → I K → 0
as claimed, where the degree 2 generator of I K is the defining form of G(1, 3).

In particular, K is ACM with Hilbert polynomial p K (t) = (9t 2 + t + 2)/2. A result similar to Proposition 2.1 holds. Proposition 3.2. The line congruence of bi-degree [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] and sectional genus 5 is the general element of the irreducible component of the Hilbert scheme Hilb p K (t) (P 5 ) containing the codimension 3, ACM closed subschemes of P 5 with Hilbert polynomial p K (t) and resolution [START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF]. Now, we go back to the geometric description of K. As P 2 is embedded both in P 4 as a Bordiga surface B, and in G(1, 3) as a suitable line congruence K, it is natural to look for maps from P 4 to G(1, 3) that transform B into K.

In [START_REF] Verra | Smooth surfaces of degree 9 in G(1, 3)[END_REF], it is proved that the map θ that makes commutative the diagram ( 7)

P 2 ψ ϕ / / B ⊂ P 4 θ x x K ⊂ G(1, 3)
is associated to the linear system |O P 4 (2) \ Γ| where Γ ⊂ B is a degree 4 rational normal curve in P 4 , and so Γ is the image in B of a line in P 2 . The map θ contracts lines that are bi-secant to Γ and so θ is an embedding of P 4 \ F where F is the secant variety to Γ, a degree 3 hypersurface. Moreover, θ maps F to a Veronese surface V . Now, we can prove the second result we are not able to quote.

Theorem 3.1. The line congruence K of bi-degree [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] and sectional genus 5 in G(1, 3) is Gorenstein linked to a Veronese surface V by an arithmetically Gorenstein surface S of degree 13 and socle degree 4.

Proof. The map θ -1 is defined on K \ C where C = K ∩ V and V is the contraction of the secant variety to Γ where θ is not an embedding. The curve C is in the linear system

|10π * L -3E 1 -• • • -3E 10 | = | -K K +
H| and so C is a twisted anticanonical divisor. By ( [START_REF] Kleppe | Gorenstein Liaison, Complete Intersection Liaison Invariants and Onubstructedness[END_REF], Lemma 5.4), C is arithmetically Gorenstein in P 5 , and its minimal free resolution can be easily constructed by self-duality, because the minimal free resolution of I V is a subcomplex of the resolution of I C . Then, we have

0 → T (-7) → T 3 (-4) ⊕ T 6 (-5) → T 8 (-3) ⊕ T 8 (-4) → T 6 (-2) ⊕ T 3 (-3) → I C → 0. Let S = K ∪ V . Then, we have the short exact sequence 0 → I S → I K ⊕ I V → I C → 0
from which we get the shape of the minimal free resolution of I S :

(8) 0 → T (-7) → T 4 (-4) ⊕ T (-5) → T (-2) ⊕ T 4 (-3) → I S → 0
and so we get that S is arithmetically Gorenstein of degree 13 and socle degree 4, as claimed. Moreover, K is algebraically linked to V by S and the linkage is geometric, because both K and V are irreducible varieties, and so they have no irreducible component in common.

Conversely, let V ⊂ G(1, 3) be a Veronese surface, and let S be an arithmetically Gorenstein surface of degree 13 and socle degree 4 containing V and contained in the Grassmannian G(1, 3). Then, the Hilbert function of the Artinian reduction of S is (1, 3, 5, 3, 1), and so its minimal free resolution is [START_REF] Bertolini | Generalized Fundamental Matrices as Grassmann Tensors[END_REF]. Hence, the residual variety K has minimal free resolution (6) and so K is a line congruence of bi-degree [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] and sectional genus 5, as claimed.

Remark 3.1. As the shape of the resolution of I C allows cancellations, the general element of | -K K + H| is not the intersection of K with a Veronese surface. Now, we describe an algorithm, based on [START_REF] Notari | A stratification of Hilbert schemes by initial ideals and applications[END_REF], to compute such a surface S, given a general line congruence K. It can be easily adapted to start from V. In last section 8, we implement it in Singular ( [START_REF] Decker | Singular 4-1-0 -A computer algebra system for polynomial computations[END_REF]), as this explicit computation is needed in section 7, which is the algorithmic part of the paper.

Given a reduced Gröbner basis of I K with respect to the degree reverse lexicographic ordering (degrevlex, for short) in T , the initial ideal in(I K ) is generated by

x 2 x 3 , x 3 0 , x 2 0 x 1 , x 0 x 2 1 , x 3 1 , x 2 0 x 2 , x 0 x 1 x 2 , x 2 1 x 2 .
With "general" line congruence, we mean that the generators of in(I K ) are as large as possible w.r.t. the degrevlex ordering and the Hilbert function of I K . Then, we write polynomials p 1 , . . . , p 5 whose coefficients are indeterminates, and whose initial terms are

x 3 0 , x 2 0 x 1 , x 0 x 2 1 , x 2 0 x 2 , x 4
1 , respectively. Also in this case, we choose the largest monomial ideal compatible with the Hilbert function of S. We want to compute the indeterminate coefficients in such a way that {x 0 x 5 -x 1 x 4 + x 2 x 3 , p 1 , p 2 , p 3 , p 4 , p 5 } is a reduced Gröbner basis of I S . At first, we reduce p 1 , . . . , p 5 w.r.t. the Gröbner basis of I K and we set equal to 0 their normal forms. This gives a linear system in the indeterminate coefficients. We compute the solutions of this system, and we substitute them in the polynomials p 1 , . . . , p 5 . Then, we run the Buchberger algorithm on the input {x 0 x 5 -x 1 x 4 +x 2 x 3 , p 1 , p 2 , p 3 , p 4 , p 5 } and we set equal to zero their S-polynomials. We get a new set of equations, no more linear. It is possible to check, however, that the equations allow us to express all but two of the remaining variables as rational functions of the last two ones. More precisely, the two special variables are the coefficients of x 2 1 x 2 , x 0 x 1 x 2 in p 4 . For a random choice of them, we compute the polynomials p 1 , . . . , p 5 and we get the arithmetically Gorenstein surface S we are looking for. As a by-product, we have that such surfaces form a dimension 2 family, whose parameter space contains an open subset of an affine plane A 2 . This confirms that for every line in P 2 , containing no point among p 1 , . . . , p 10 , we can construct a rational normal curve Γ in the Bordiga surface B ⊂ P 4 and a curve C contained in K ⊂ G(1, 3) where the map θ is not an embedding.

As last result, we describe a geometric property of K. We proved in section 2 that a plane cubic curve containing 9 of the 10 points in Z is mapped onto a plane cubic curve in B, as well. Now we prove that these plane curves are mapped by θ to plane curves again. Take, for example, an element in

|3π * L-E 2 -• • •-E 10 |.
The degree of its image in K is 3 and its genus, computed via adjunction, turns out to be 1. So it is a plane curve in K, too. The plane containing this curve is an α-plane in G(1, 3). To compute the equations of the α-plane, one can proceed as follows. Let C i ⊂ P 2 be the cubic curve not containing p i , i = 1, . . . , 10, and let ij be the line through p i , p j , with i = j. Then, ij C i C j is a degree 7 plane curve singular at p 1 , . . . , p 10 , and so it is a linear combination of a basis of the linear system. These linear combinations give the equations of the hyperplanes in G(1, 3) ⊂ P 5 containing the image of C i . If we select p 1 , p 2 , p 3 as distinguished points among p 1 , . . . , p 10 , and we set α 1 , α 2 , α 3 the planes containing the images of C 1 , C 2 , C 3 , respectively, then each one of them intersects the other two ones, as the intersection point is image of the further intersection point of C i and C j other then the base points. Now, we can prove the last result we are not able to quote.

Proposition 3.3. With the previous notations, let Y be the union of the line congruence K and the three planes α 1 , α 2 , α 3 . Then, Y is an ACM surface of bi-degree [START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] in G(1, 3) and the minimal free resolution of its defining ideal is

(9) 0 → T 3 (-6) → T 3 (-4) ⊕ T 4 (-5) → T (-2) ⊕ T 4 (-3) → I Y → 0.
Proof. To prove the result, we use mainly mapping cone procedure. The starting point is the following short exact sequence

0 → I K∩(α 1 ∪α 2 ) → I K∩α 1 ⊕ I K∩α 2 → I K∩α 1 ∩α 2 → 0.
As K ∩ α i is a degree 3 plane curve, the minimal free resolution of I K∩α i is Koszul, with 3 generators in degree 1 and a further generator in degree 3. On the other hand, K ∩ α 1 ∩ α 2 is a point, and so also the minimal free resolution of I K∩α 1 ∩α 2 is Koszul with 5 generators in degree 1. Furthermore, K ∩ (α 1 ∪ α 2 ) is the union of two plane cubic curves meeting at a point, and so there are two minimal generators in degree 3.

Then, by mapping cone on the previous sequence, we get 0 →

T ( -5) ⊕ T 2 (-6) → T 5 (-4) ⊕ T 6 (-5) → T 8 (-3) ⊕ T 6 (-4) → T (-1) ⊕ T 4 (-2) ⊕ T 2 (-3) → I K∩(α 1 ∪α 2 ) → 0.
Before going on, we construct the minimal free resolution of I α 1 ∪α 2 ∪α 3 . We choose coordinates in P 5 in such a way that I α 1 = x 0 , x 1 , x 2 , I α 2 = x 0 , x 3 , x 4 , I α 3 = x 1 , x 3 , x 5 , so to let them intersect pairwise. Hence, I α 1 ∪α 2 ∪α 3 = x 0 x 1 , x 0 x 3 , x 0 x 5 , x 1 x 3 , x 1 , x 4 , x 2 x 3 , x 2 x 4 x 5 , and its minimal free resolution has the shape

0 → T (-6) → T 3 (-4) ⊕ T 3 (-5) → T 8 (-3) ⊕ T 3 (-4) → T 6 (-2) ⊕ T (-3) → I α 1 ∪α 2 ∪α 3 → 0.
Now, we consider the short exact sequence

0 → I K∩(α 1 ∪α 2 ∪α 3 ) → I K∩(α 1 ∪α 2 ) ⊕ I K∩α 3 → I K∩(α 1 ∪α 2 )∩α 3 → 0.
When we apply the mapping cone procedure, we consider that, by degree argument, the minimal free resolution of I α 1 ∪α 2 ∪α 3 is a sub-complex of the resolution of I K∩(α 1 ∪α 2 ∪α 3 ) , and that K ∩ (α 1 ∪ α 2 ) ∩ α 3 is a set of 2 points, and so its minimal free resolution is Koszul, too, with 4 generators in degree 1 and a generator in degree 2. Then, we get

0 → T 4 (-6) → T 3 (-4) ⊕ T 12 (-5) → T 8 (-3) ⊕ T 12 (-4) → T 6 (-2) ⊕ T 4 (-3) → I K∩(α 1 ∪α 2 ∪α 3 ) → 0.
Finally, from the short exact sequence

0 → I Y → I K ⊕ I α 1 ∪α 2 ∪α 3 → I K∩(α 1 ∪α 2 ∪α 3 ) → 0
we get the result on the minimal free resolution of I Y . From the minimal free resolution, it follows that Y is ACM. 4. On multiview Geometry for projections from P 4 to P 2

In this section we fix notation and terminology and give a short overview of classical facts in Computer Vision related to the problem of projective reconstruction of scenes and cameras from multiple views.

Even if all the definitions can be given in full generality, we restrict ourselves to the case of projections from P 4 to P 2 . In this context, a camera P is a linear projection from P 4 onto P 2 , from a line C P , called center of projection. The target space P 2 is called view. A scene is a set of points X i ∈ P 4 .

Using homogeneous coordinates in P 4 and P 2 , we identify P with a 3 × 5 matrix of maximal rank, defined up to a multiplicative constant. Hence C P comes out to be the right annihilator of P.

Let us consider a set of cameras P j : P 4 \ C P j → P 2 projecting the same scene in P 4 and the corresponding set of images in the different target planes. In this setting, proper linear subspaces (points or lines), L i , i = 1 . . . m, of different views are said to be corresponding if there exists at least a point X ∈ P 4 such that P i (X) ∈ L i for all i = 1 . . . m.

In the context of multiple view geometry, the problem of projective reconstruction of a scene, given multiple images of it, is the following: given many enough scene points in P 4 and identified a suitable number of corresponding subspaces on each image, one wants to get the projection matrices (up to projective transformations), i.e. the cameras, and the coordinates in P 4 of the scene points.

Hartley and Schaffalitzky, [START_REF] Hartley | Reconstruction from projections using Grassmann tensors[END_REF], have constructed a set of multiview tensors, called Grassmann tensors, encoding the relations between sets of corresponding subspaces. We recall here the basic elements of their construction in the case of three projections from P 4 to P 2 , which turns out to be the minimal number of views allowing projective reconstruction.

Consider three projections P j : P 4 \ C j → P 2 , j = 1, 2, 3, with centers C 1 , C 2 , C 3 in general position.

Let {L 1 , L 2 , L 3 } be three general linear subspaces (points or lines) of P 2 of codimension α 1 , α 2 , α 3 , respectively. (L 1 , L 2 , L 3 ) is a triple of corresponding subspaces if and only if (P 1 ) -1 (L 1 ) ∩ (P 2 ) -1 (L 2 ) ∩ (P 3 ) -1 (L 3 ) is not empty. From the Grassmann formula, if α j = 5, j = 1, 2, 3, the existence of points in the previous intersection gives a constrain which allows us to construct the Grassmann tensor. Hartley and Schaffalitzky call the triple (α 1 , α 2 , α 3 ) a profile for the reconstruction problem. In our situation the only possible profiles are: (α 1 , α 2 , α 3 ) = (2, 2, 1), (2, 1, 2) or (1, 2, 2).

Let {L 1 , L 2 , L 3 } be three general linear subspaces of P 2 as above and let S j be the maximal rank matrix of type 3 × (3 -α j ) whose columns are a basis for L j , j = 1, 2, 3. By definition, if the L j 's are corresponding subspaces, there exists a point X ∈ P k such that P j (X) ∈ L j for j = 1, 2, 3. In other words there exist three vectors v j ∈ C 3-α j j = 1, 2, 3 such that:

(10)   S 1 0 0 P 1 0 S 2 0 P 2 0 0 S 3 P 3   •     v 1 v 2 v 3 X     =   0 0 0   .
The existence of a non-trivial solution (v 1 , v 2 , v 3 , X) of system [START_REF] Bertolini | Critical hypersurfaces and instability for reconstruction of scenes in high dimensional projective spaces[END_REF] implies that the coefficient matrix has determinant zero, as it is square of order 9 in our case. This determinant can be thought of as a tri-linear form (tensor) in the Plücker coordinates of the spaces L j . This tensor is the Grassmann tensor.

In the following we explicitly construct such a tensor for the profile (2, 2, 1), others being similar. Now L 1 , L 2 are points and L 3 is a line. We denote by (x 1 , x 2 , x 3 ), (y 1 , y 2 , y 3 ) the homogeneous coordinates of L 1 and L 2 , respectively, and by (z 1 , z 2 , z 3 ) and (w 1 , w 2 , w 3 ) the homogeneous coordinates of two points of L 3 . In this case the matrix of the coefficients of the linear system (10) becomes:

T P 1 ,P 2 ,P 3 L 1 ,L 2 ,L 3 =              x 1 0 0 0 P 1 [1, 1] P 1 [1, 2] P 1 [1, 3] P 1 [1, 4] P 1 [1, 5] x 2 0 0 0 P 1 [2, 1] P 1 [2, 2] P 1 [2, 3] P 1 [2, 4] P 1 [2, 5] x 3 0 0 0 P 1 [3, 1] P 1 [3, 2] P 1 [3, 3] P 1 [3, 4] P 1 [3, 5] 0 y 1 0 0 P 2 [1, 1] P 2 [1, 2] P 2 [1, 3] P 2 [1, 4] P 2 [1, 5] 0 y 2 0 0 P 2 [2, 1] P 2 [2, 2] P 2 [2, 3] P 2 [2, 4] P 2 [2, 5] 0 y 3 0 0 P 2 [3, 1] P 2 [3, 2] P 2 [3, 3] P 2 [3, 4] P 2 [3, 5] 0 0 z 1 w 1 P 3 [1, 1] P 3 [1, 2] P 3 [1, 3] P 3 [1, 4] P 3 [1, 5] 0 0 z 2 w 2 P 3 [2, 1] P 3 [2, 2] P 3 [2, 3] P 3 [2, 4] P 3 [2, 5] 0 0 z 3 w 3 P 3 [3, 1] P 3 [3, 2] P 3 [3, 3] P 3 [3, 4] P 3 [3, 5]             
where P i [j, k] denotes the element in position (j, k) of the matrix P i .

If L 1 , L 2 and L 3 are corresponding spaces then the linear system (11)

T P 1 ,P 2 ,P 3 L 1 ,L 2 ,L 3       λ µ α β X       = 0
has a non trivial solution, and so det(T P 1 ,P 2 ,P 3 L 1 ,L 2 ,L 3 ) = 0. The converse is true for general L 1 , L 2 and L 3 since we are looking for a non trivial solution of [START_REF] Bertolini | Critical configurations for 1-view in projections from P k → P 2[END_REF] in which X is a point of P 4 (i.e. X = 0) and

X / ∈ C 1 ∪ C 2 ∪ C 3 .
In particular this happens if

L 1 / ∈ P 1 (C 2 ∪ C 3 ) and L 2 / ∈ P 2 (C 1 ∪ C 3 )
. Under this hypothesis, if det(T P 1 ,P 2 ,P 3 L 1 ,L 2 ,L 3 ) = 0, then L 1 , L 2 and L 3 are corresponding as the linear system (11) has a non trivial solution with X as required. Indeed the case X = 0 doesn't occur, since, in the opposite case, either L 1 or L 2 are not points, or L 3 is not a line. Moreover X / ∈ C 1 ∪ C 2 ∪ C 3 , for the assumption.

In conclusion, for the chosen profile (2, 2, 1), one sees that det(T P 1 ,P 2 ,P 3 L 1 ,L 2 ,L 3 ) = 0 is indeed the tri-linear constraint between the coordinates x and y of the first and second view and the Plücker (i.e.dual) coordinates of the line < z, w > of the third view so to let them be corresponding.

Critical loci for projective reconstruction of points and their geometry

As discussed in the previous section, folklore on the reconstruction problem says that sufficiently many views and sufficiently many sets of corresponding points in the given views should allow one a successful projective reconstruction. This is generally true, but even in the classical set-up of two projections from P 3 to P 2 one can have non projectively equivalent pairs of sets of scene points and of cameras that produce the same images in the view planes, thus preventing reconstruction. Such configurations and the loci they describe are referred to as critical. In [START_REF] Bertolini | Critical loci for projective reconstruction from multiple views in higher dimension: A comprehensive theoretical approach[END_REF], critical loci for projective reconstruction of camera centers and scene points from multiple views for projections from P k to P 2 have been introduced and studied. Here we shortly recall the basic definitions in the case in which we are interested in, i.e. three views from P 4 to P 2 . Moreover we study the critical locus X for general projections. The non general case is under investigation, and some preliminary results are summarized in [START_REF] Bertolini | Provisional title: Critical loci for 3-view reconstruction in P 4 : the degenerate cases and instability[END_REF]. Definition 5.1. A set of points {X j }, j = 1, . . . , N, N 0, in P 4 is said to be a critical configuration for projective reconstruction from three views if there exist two collections of 3 × 5 full-rank projection matrices P i and Q i , i = 1, 2, 3, and a set of N points {Y j } ⊂ P 4 , non-projectively equivalent to {X j }, such that, for all i and j, P i (X j ) = Q i (Y j ), up to homography in the image planes. The two sets {X j } and {Y j } are called conjugate critical configurations, with associated conjugate matrices {P i } and {Q i }.

Remark 5.1. It can be proved that in the above definition N ≥ 7 is enough.

The generators of the ideal of the critical locus X can be obtained by making use of the Grassmann tensor introduced in the previous section.

Indeed, the Grassmann tensor T P 1 ,P 2 ,P 3 encodes the algebraic relations between corresponding subspaces in the different views of the projections P 1 , P 2 , P 3 . Hence by definition of critical set, if {X j , Y j } are conjugate critical configurations, then, for each j, the projections P 1 (X j ), P 2 (X j ) and P 3 (X j ) are corresponding points not only for the projections P 1 , P 2 , P 3 , but for the projections

Q 1 , Q 2 , Q 3 , too.
Following the construction of the previous section, we first choose the profile (2, 2, 1). If L 1 , L 2 , L 3 is a triple of corresponding spaces for the projections Q 1 , Q 2 , Q 3 , where L 1 , L 2 are points with homogeneous coordinates L 1 = (x 1 , x 2 , x 3 ) T , L 2 = (y 1 , y 2 , y 3 ) T in the first two views respectively, and L 3 is the line spanned by z = (z 1 , z 2 , z 3 ) and w = (w 1 , w 2 , w 3 ) in the third view, the trilinear relation between L 1 , L 2 , L 3 is given by the vanishing of det(T

Q 1 ,Q 2 ,Q 3 L 1 ,L 2 ,L 3
), where

T Q 1 ,Q 2 ,Q 3 L 1 ,L 2 ,L 3 =   L 1 0 0 0 Q 1 0 L 2 0 0 Q 2 0 0 z w Q 3   .
Considering as corresponding spaces L 1 = P 1 (X), L 2 = P 2 (X) and any line L 3 passing through P 3 (X), with X any point in the critical locus, one gets that the determinant of the matrix

M =   P 1 (X) 0 0 0 Q 1 0 P 2 (X) 0 0 Q 2 0 0 P 3 (X) v Q 3   must vanish for every choice of v = (a, b, c) T .
This implies that all the cofactors of the elements a, b, c in M must vanish. Since the above reasoning holds for all the possible profiles (2, 2, 1), (2, 1, 2) or (1, 2, 2), one sees that X is in the critical locus if and only if one has the vanishing of all the maximal minors of

M =   P 1 (X) 0 0 Q 1 0 P 2 (X) 0 Q 2 0 0 P 3 (X) Q 3   .
From the above discussion it follows that the maximal minors of M generate the ideal of the critical locus X , as X has to satisfy no other constraint. Remark 5.2. In [START_REF] Bertolini | Critical loci for projective reconstruction from multiple views in higher dimension: A comprehensive theoretical approach[END_REF], the critical locus for a suitable number n of projections from P k to P 2 has been introduced and studied in a more general context. If k is even, the critical locus is a determinantal variety, defined by 3n polynomials of degree n. In the particular case of three projections from P 4 to P 2 , up to a constant, these polynomials are exactly the maximal minors of M defined above. Now we study the geometry of the critical locus X in a general situation, where general means that the greatest common divisor of the generators of the ideal is trivial.

Since the generators have a trivial greatest common divisor, by Hilbert-Burch Theorem ( [START_REF] Eisenbud | Commutative Algebra[END_REF], Theorem 20.15), a free resolution of the ideal I X they generate is

(12) 0 → R 3 (-4) ⊕ R 5 (-3) M -→ R 9 (-3) -→ I X → 0
where M is the above matrix.

By generality assumption, we can assume that rank

  Q 1 Q 2 Q 3   = 5
. Hence, we can cancel the summand R 5 (-3) from the resolution, and we get the minimal free resolution [START_REF] Buchanan | The twisted cubic and camera calibration[END_REF] 0 → R 3 (-4)

N X -→ R 4 (-3) -→ I X → 0.
Let us assume that a non-vanishing minor is given by the last 5 rows and columns in M . Then if we write M as the following block matrix

M = A B C D
with A of type 4 × 3, and D invertible, we can reduce M with elementary operations on rows and columns to the block matrix N X 0 0 I 5 where I 5 is the 5 × 5 identity matrix, 0 are null matrices of suitable type, and

N X = A -BD -1 C.
Because of the way it is obtained, N X has linear entries. As the elementary operations on the columns correspond to base changes in the free module R 3 (-4) ⊕ R 5 (-3) while the ones on the rows correspond to base changes in the free module R 9 (-3), the global effect is to construct a new set of 4 generators of the ideal I X again as maximal minors of N X . Then, the critical locus X is a codimension 2 arithmetically Cohen-Macaulay scheme in P 4 with Hilbert polynomial p(t) = (6t 2 + 2t + 2)/2 = p B (t). From Proposition 2.1, we have the following Proposition 5.1. The general critical locus for projective reconstruction for three views from P 4 to P 2 is in the irreducible component of Hilb p B (t) (P 4 ) containing the Bordiga surfaces.

It is then natural to characterize the locus filled by critical loci inside this irreducible component. Quite surprisingly, this locus coincides with the whole irreducible component. In fact, we have Theorem 5.1. Let B be a Bordiga surface. Then B is a critical locus for the projective reconstruction from three views from P 4 to P 2 , that is to say, there exist two couples of three projections P 1 , P 2 , P 3 and Q 1 , Q 2 , Q 3 from P 4 to P 2 such that the associated critical locus is B.

Proof. Let Z ⊂ P 2 be a set of 10 points in uniform position and let B be the associated Bordiga surface. B is irreducible, of course. We choose a reference frame in P 2 in such a way that the fundamental points (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1) are in Z. For this choice, from equation ( 3), we have that the columns of N B in resolution (2) define three lines in P 4 that lie on the Bordiga surface B. We take these lines as centers of three projections P 1 , P 2 , P 3 .

By performing elementary operations on the rows of N B = (n ij ), we can assume that n 41 = 0 and n 11 , n 21 , n 31 are linearly independent. Now, we want to prove that n 42 = 0. Assume by contradiction that n 42 = 0. Then, with easy computations, we have that I B is generated by n 43 det(N 14;3 ), n 43 det(N 24;3 ), n 43 det(N 34;3 ), det(N 4).

So,

I B = n 43 , det(N 4) ∩ det(N 14;3 ), det(N 24;3 ), det(N 34;3 )

where ij; k denotes the cancellation of the rows i, j and of the column k from N B , while i denotes the cancellation of the i-th row, and hence B is not irreducible. In more details, n 43 , det(N 4) defines a cubic surface in a P 3 while det(N 14;3 ), det(N 24;3 ), det(N 34;3 ) defines a cone in P 4 over a twisted cubic curve in a suitable P 3 . Finally, the two surfaces meet along a twisted cubic curve. Then, n 42 = 0.

Analogously we get n 43 = 0. Performing other elementary operations on the rows of N B , we can get n 32 = 0. In analogy to the previous case, n 33 = 0 otherwise B would be non irreducible, as only an element in the third row of N B is non-zero. Let p 1 3 X, p 2 3 X, p 3 3 X be a basis of the ideal generated by n i3 , i = 1, . . . , 4. Moreover, let E be the 4 × 3 matrix such that

    n 13 n 23 n 33 n 43     = E   p 1 3 X p 2 3 X p 3 3 X   .
As n 33 = 0 and n 43 = 0, the last two rows of E are non-zero, as well. Then, if we set

A =     n 11 0 0 n 21 0 0 n 31 0 0 0 n 42 0     and C =       0 n 12 0 0 n 22 0 0 0 p 1 3 X 0 0 p 2 3 X 0 0 p 3 3 X       we have that N B = A -F C for F =     -1 0 0 -1 0 0 0 0 -E      .
Let D be any invertible matrix of order 5 and take B = F D. Then, the maximal minor of the matrix M = A B C D generate the same ideal I B that defines the Bordiga surface B from which we started.

The matrices P 1 X, P 2 X, P 3 X are the three matrices deduced by taking the rows of A C three by three.

Let Q 1 , Q 2 , Q 3 the three matrices obtained by taking the rows of B D three by three.

We want to check that Q i has maximal rank 3. This is obvious for Q 3 because the rows of D are linearly independent. Q 1 has rank three too, because the first three rows of F are linearly independent and D is invertible. The last row of B is the last row of F times D, and so it is a linear combination of the last three rows of D with non-zero coefficients. So, it is not a linear combination of the first two rows of D, because D has rank 5. Hence, Q 2 has maximal rank 3, too, and the proof is complete.

Critical loci for projective reconstruction of lines and their geometry

In the previous sections, we have dealt with the reconstruction problem for sets of points from their images via three projections from P 4 to P 2 . Moreover we have studied the critical loci for this problem.

A different reconstruction problem arises when one considers projections of lines in P 3 instead of projections of points. This set-up has been considered and studied by various authors, in particular T.Buchanan [START_REF] Buchanan | On the critical set for photogrammetric reconstruction using line tokens in P 3 C[END_REF] and S.J.Maybank [START_REF] Maybank | The critical line congruence for reconstruction from three images[END_REF]. Given a set of lines in P 3 and n projections of these lines to P 2 , T.Buchanan [START_REF] Buchanan | On the critical set for photogrammetric reconstruction using line tokens in P 3 C[END_REF] shows that n = 3 is the minimum n such that it is possible to reconstruct the set from their images, up to projective transformations in P 3 . Of course, also in this context, there is a natural notion of critical locus, consisting of lines in P 3 . Definition 6.1. A set of lines {λ j }, j = 1, . . . , N, N 0, in P 3 is said to be a critical configuration for projective reconstruction of lines from three views if there exist two collections of 3 × 4 full-rank projection matrices ϕ i and ψ i , i = 1, 2, 3, and a set of N lines {µ j } in P 3 , non-projectively equivalent to {λ j }, such that, for all i and j, ϕ i (λ j ) = ψ i (µ j ), up to homography in the (dual) image planes. The two sets {λ j } and {µ j } are called conjugate critical configurations, with associated conjugate matrices {ϕ i } and {ψ i }.

In [START_REF] Buchanan | On the critical set for photogrammetric reconstruction using line tokens in P 3 C[END_REF], the author, using geometrical arguments, shows that given two triples of projections from P 3 to P 2 , the associated critical set for reconstruction via lines is a congruence K of bi-degree [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] and sectional genus 5 in the Grassmannian G(1, 3) (see section 3).

In this section, via an algebraic approach, we compute the defining ideal of the critical locus for this reconstruction problem. We prove that the critical locus is the union of the above line congruence and of the three α-planes in G(1, 3) corresponding to the three centers of projection in P 3 .

To start, we denote by ϕ i : P 3 → P 2 and ψ i : P 3 → P 2 , i = 1, 2, 3, two triples of projections and we consider the associated critical locus for the reconstruction problem through lines considered by T.Buchanan [START_REF] Buchanan | On the critical set for photogrammetric reconstruction using line tokens in P 3 C[END_REF]. Let O i ∈ P 3 (respectively, O i ∈ P 3 ) be the center of the projection ϕ i (resp. ψ i ). As in the following we are interested in the rows of the matrices ψ 1 , ψ 2 , ψ 3 , we set

(14)   ψ 1 ψ 2 ψ 3   =   R 1 . . . R 9  
where R i is a 1 × 4 matrix. As we consider general projections, we assume that the above matrix has rank 4. As proved in [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF], if we call l a line in P 3 and l its image in P 2 via a projection ϕ = (ϕ ij ), the Plücker coordinates of l are transformed into the Plücker coordinates of l via

ϕ ∧ ϕ =             ϕ 21 ϕ
            Hence, we denote Φ i = ϕ i ∧ ϕ i : P 5 → P 2 and similarly Ψ i = ψ i ∧ ψ i both for i = 1, 2, 3.
Remark 6.1. The center of the projection Φ i is the 2-plane in P 5 defined by Φ i (X) = 0. They are contained in the Grassmannian G(1, 3) and parameterize the lines of P 3 through O i , for every i, i.e. they are α-planes.

In a natural way, then, we are induced to consider the reconstruction problem for points in P 5 from three views, which turns out to be the minimum number needed for the reconstruction of a scene. Let X = (x 0 , . . . , x 5 ) T be a point in P 5 . As done in section 5 while computing the critical locus for two triples of projections from P 4 to P 2 , one has to consider the 9 × 9 matrix

M =   Φ 1 (X) 0 0 Ψ 1 0 Φ 2 (X) 0 Ψ 2 0 0 Φ 3 (X) Ψ 3  
where 0 is the 3 × 1 null matrix.

Remark 6.2. In the case of two triples of projections from P 5 to P 2 , det(M ) = 0 is the defining equation of the hypersurface that is critical for the reconstruction for points in this setting. Indeed, det(M ) = 0 is equivalent to require that the linear system

M     λ 1 λ 2 λ 3 Y     = 0 has a non-trivial solution such that Y = 0.
The condition det(M ) = 0 is not enough in our case to describe the critical locus since both X and Y are lines and so their coordinates must satisfy the equation of Klein hyperquadric. For X we have x 0 x 5 -x 1 x 4 +x 2 x 3 = 0. The situation for Y is more involved. Indeed we have first to compute the coordinates of Y by solving suitable homogeneous linear systems, then to impose that such coordinates satisfy the Klein equation.

If we denote by M î the matrix obtained from M by erasing the i-th row, the homogeneous linear system becomes

M î     λ 1 λ 2 λ 3 Y     = 0 1 ≤ i ≤ 9.
It follows from Cramer's rule that

y j = (-1) 1+j det(M i,j+4 ), j = 0 . . . 5,
where M h,k is the matrix obtained from M by erasing the h-th row and the k-th column. Now we have to impose that the computed y 0 . . . y 5 satisfy y 0 y 5 -y 1 y 4 + y 2 y 3 = 0.

To simplify notations, we set

(15) Φ i (X) =   Φ i1 Φ i2 Φ i3  
that are the natural variables to construct the generators of the ideal of the critical locus Y. In fact, while det(M ) is a degree 3 form in the Φ ij 's, irreducible in the general case, the form y 0 y 5 -y 1 y 4 + y 2 y 3 has degree 6 and it is the product of three linear forms and a cubic form in the Φ ij 's. To fix notations we denote by g i the cubic arising from the linear system associated to M î. For example, let us consider the linear system associated to M 1.

In this case, the linear forms are

Φ 21 det     R 7 R 8 R 9 R 4     + Φ 22 det     R 7 R 8 R 9 R 5     + Φ 23 det     R 7 R 8 R 9 R 6     = 0, Φ 31 det     R 4 R 5 R 6 R 7     + Φ 32 det     R 4 R 5 R 6 R 8     + Φ 33 det     R 4 R 5 R 6 R 9     = 0, Φ 12

and

(det((ψ 3 ) 1), det((ψ 3 ) 2), det((ψ 3 ) 3), det((ψ 3 ) 4)) ,

where, this time, ĥ means that we erase the h-the column.

We have then proved the following:

Theorem 6.1. The defining ideal I Y of the critical locus is generated by the Klein hyperquadric, det(M ) and g 1 . . . , g 9 .

Now we compute the minimal generators and free resolution of the ideal I Y . It is straightforward to check that

g i = 1,2,3 j,h,k Φ 1j Φ 2h Φ 3k det     R i R j R 3+h R 6+k     i = 1, . . . , 9 while det(M ) = - 1,2,3 i,j,h Φ 1i Φ 2j Φ 3h c i,3+j,6+h
where

Φ ijh =   R i R j R h   and c i,j,h = det     det((Φ 1 ) 1) det((Φ 1 ) 2) det((Φ 1 ) 3) det((Φ 1 ) 4) det((Φ 2 ) 1) det((Φ 2 ) 2) det((Φ 2 ) 3) det((Φ 2 ) 4) det((Φ 3 ) 1) det((Φ 3 ) 2) det((Φ 3 ) 3) det((Φ 3 ) 4) det((Φ ijh ) 1) det((Φ ijh ) 2) det((Φ ijh ) 3) det((Φ ijh ) 4)     .
Let J be the ideal generated by g 1 , . . . , g 9 , det(M ). Then, Proposition 6.1. The minimal free resolution of J is 0 → T 3 (-4) → T 4 (-3) → J → 0 and so the scheme defined by J is ACM of codimension 2 in P 5 .

Proof. Without loss of generality, we assume that R 1 , R 2 , R 4 , R 7 are linearly independent, and so they are a basis of C 4 . Moreover, we set

R i = α i1 R 1 + α i2 R 2 + α i3 R 4 + α i4 R 7 i = 3, 5 , 6, 8, 9 and
D = det     R 1 R 2 R 4 R 7     .
We remark that D is the coefficient of Φ

    R 5 R 3 R 6 R 7     = det     α 51 R 1 + α 52 R 2 + α 53 R 4 α 31 R 1 + α 32 R 2 + α 33 R 4 α 61 R 1 + α 62 R 2 + α 63 R 4 R 7     = D det   α 51 α 52 α 53 α 31 α 32 α 33 α 61 α 62 α 63   .
On the right, we have

- α 31 α 54 -α 34 α 51 α 34 det     R 1 R 3 R 6 R 7     - α 32 α 54 -α 34 α 52 α 34 det     R 2 R 3 R 6 R 7     + α 54 α 34 det     R 3 R 3 R 6 R 7     - - α 33 α 54 -α 34 α 53 α 34 det     R 4 R 3 R 6 R 7     = - α 31 α 54 -α 34 α 51 α 34 det     R 1 α 32 R 2 + α 33 R 4 α 62 R 2 + α 63 R 4 R 7     - - α 32 α 54 -α 34 α 52 α 34 det     R 2 α 31 R 1 + α 33 R 4 α 61 R 1 + α 63 R 4 R 7     - α 33 α 54 -α 34 α 53 α 34 det     R 4 α 31 R 1 + α 32 R 2 α 61 R 1 + α 62 R 2 R 7     = =D -
    R 1 R 1 R 4 R 7     - 1 α 34 D c 134 det     R 2 R 1 R 4 R 7     + 1 α 34 D det     R 3 R 1 R 4 R 7     = = 1 α 34 c 134 + 1 α 34 D det     α 32 R 2 R 1 R 4 R 7     = 1 α 34 det     det((Φ 1 ) 1) . . . det((Φ 2 ) 1)
. . . det((Φ 3 ) 1)

. . . det((Φ 134 -α 32 Φ 124 ) 1) . . .

    = = 1 α 34 det     det((Φ 1 ) 1) . . . det((Φ 2 ) 1)
. . . det((Φ 3 ) 1)

. . . det((α 34 Φ 174 ) 1) . . .

    = -c 147
and so the equality holds also in this case.

As second step, we check that the minors with sign of the following matrix N are equal to g 1 , . . . , g 4 .

D

= det     R 4 R 1 R 5 R 9    
and so we get det(N 4) = g 4 . With analogous computations, we get det(N 1) = -g 1 , det(N 2) = g 2 , and det(N 3) = -g 3 . As g 1 , . . . , g 4 are irreducible for general choices, we get the claim by Hilbert-Burch Theorem. Moreover, N represents the map T 3 (-4) → T 4 (-3) in the resolution of J.

Remark 6.3. The scheme defined by J in P 5 is a lifting of a Bordiga surface in P 4 in the following sense. Let be a general linear form in T. Hence there is an isomorphism between R and T / . As J is ACM the minimal free resolution of J remains exact over T / and so (J + )/ defines a Bordiga surface in P 4 = Proj(R). Moreover, the matrices N and N X of the two minimal free resolutions have the same properties: the three columns span linear spaces of codimension 3, in position (4, 1) there is a zero, and the element in position (3, 2) is a linear combination of the ones in positions (1, 2) and (2, 2). The main difference is that the linear spaces spanned by the columns are pairwise skew in the case of the Bordiga surface, while every couple of them has a point in common in the case of J, as the codimension 2 linear spaces are α-planes in this case, and so the intersection of two of them is the point corresponding to the line through the centers of the two nets of lines.

Now we can describe the geometry of the critical locus, starting from its defining ideal, and Buchanan's result. Theorem 6.2. The critical locus for the reconstruction problem for a pair of three projections ϕ i , ψ j , i, j = 1, 2, 3, of lines from P 3 to P 2 is the union of a line congruence of bi-degree [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] and sectional genus 5 and the three α-planes associated to the projection centers of the ϕ i 's. Moreover, each α-plane intersects the line congruence along a degree 3 plane curve.

Proof. The scheme defined by J is irreducible for general choices, and so the Klein hyperquadric is regular with respect to it. As the defining ideal of the critical locus Y is

I Y = J + (x 0 x 5 -x 1 x 4 + x 2 x 3 )T , then its minimal free resolution is 0 → T 3 (-6) → T 4 (-5) ⊕ T 3 (-4) → T (-2) ⊕ T 4 (-3) → I Y → 0
and so the Hilbert polynomial of Y is p Y (t) = (12t 2 -8t + 8)/2. Hence, Y is a surface in P 5 of degree 12. It is an easy check to verify that the centers α 1 , α 2 , α 3 of the projections Φ 1 , Φ 2 , Φ 3 are the α-planes associated to the centers of ϕ 1 , ϕ 2 , ϕ 3 respectively and are contained in Y, as every maximal minor of N is a combination of Φ i1 , . . . , Φ i3 , for i = 1, 2, 3. Moreover, we know that a line congruence K of bi-degree [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] and sectional genus 5 is contained in the critical locus. Then, by degree argument, Y = K ∪ α 1 ∪ α 2 ∪ α 3 . By using the same argument as in Proposition 3.3, we get that each α i intersects K along a degree 3 plane curve, as claimed.

A bridge between the two reconstruction problems

In this section, we want to show that the two reconstruction problems considered in the previous sections are related each other, in the following sense. Given two triples of projections from P 4 to P 2 , and the corresponding critical locus X ⊂ P 4 , it is possible to determine two triples of projections from P 3 to P 2 in such a way that the critical locus for the reconstruction problem for lines is the union of three suitable α-planes and the image of X in G(1, 3) via the rational map θ : P 4 → G(1, 3) quoted in [START_REF] Bertolini | Critical loci for projective reconstruction from multiple views in higher dimension: A comprehensive theoretical approach[END_REF]. Furthermore, also the converse holds. Now we describe various steps to get the ϕ i and ψ j from the P i and Q j .

(1) Let P i , Q j : P 4 P 2 be projections, with i, j = 1, 2, 3. The critical locus X ⊂ P 4 for the reconstruction problem for points has been studied in section 5. In particular, we proved that X is a Bordiga surface, and we computed the 4 × 3 matrix N X whose columns generate the first syzygy module of I X from the 9 × 8 matrix M whose maximal minors generate the same ideal I X .

(2) As explained at the end of section 2, a degree 4 rational normal curve Γ ⊂ X can be obtained by taking the maximal minors q 0 , . . . , q 5 of N X M L where M L is a general 3 × 2 matrix of maximal rank. To fix the order, q 0 = det(N X M L ) 34 , q 1 = det(N X M L ) 24 , q 2 = det(N X M L ) 14 , q 3 = det(N X M L ) 23 , q 4 = det(N X M L ) 13 , q 5 = det(N X M L ) 12 , where, as usual, .. means that the corresponding rows are omitted. (3) Following [START_REF] Verra | Smooth surfaces of degree 9 in G(1, 3)[END_REF], we define the map θ : T = H 0 * (O P 5 ) → R = H 0 * (O P 4 ) by setting θ(x i ) = q i , i = 0, . . . , 5. Then, θ -1 (I X ) = I K where K is a line congruence of bi-degree [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] and sectional genus 5 as the ones studied in section 3. (4) From equation (3), we get N Z from N X where the columns of N Z generate the first syzygy module of I Z in S = H 0 * (O P 2 ), Z being a set of 10 points in general position, among which there are p 1 (1 : 0 : 0), p 2 (0 : 1 : 0), p 3 (0 : 0 : 1). We remark that X is the image of the embedding of P 2 , blow-up of P 2 in Z, via the linear system |4π * L -E 1 -• • • -E 10 |. (5) Let C i ⊂ P 2 be the plane cubic curve containing Z \ {p i }, with i = 1, 2, 3. The equation of C i is the only degree 3 generator of I Z : I p i . Let C i ⊂ X be the image of C i and C i ⊂ K be the image of C i . The saturated ideal that defines C i is θ -1 (I C i ) and so we can compute the plane α i spanned by C i . The critical locus we are looking for is Y = K ∪ α 1 ∪ α 2 ∪ α 3 . (6) The last map in the minimal free resolution of I Y is T 3 (-6) → T 4 (-5) ⊕ T 3 (-4).

Let N Y be the matrix that represents T 3 (-6) → T 4 (-5): the maximal minors of this matrix define the ideal J and so we can recover the projection matrices Φ i , Ψ j from it, i, j = 1, 2, 3, as explained in Proposition 6.1. Hence, it is possible to compute the projections ϕ i , ψ j : P 3 → P 2 we want to construct.

(4) Compute the generators of the first syzygy module N X of I X and compute a basis from which it is possible to reconstruct the projections P i , Q j : P 4 → P 2 , 1 ≤ i, j ≤ 3, we are looking for.

A computer session concerning the line congruence

In this section, we present a computer session in Singular in which, staring from the ideal of a general line congruence K of bi-degree [START_REF] Bertolini | Instability of projective reconstruction of dynamic scenes near critical configurations[END_REF][START_REF] Bertolini | Applications Of Multiview Tensors In Higher Dimensions in "Tensors in Image Processing and Computer Vision[END_REF] and sectional genus 5, we compute a degree 13 arithmetically Gorenstein surface S such that S = K ∪ V where V is a Veronese surface, and K ∩ V is a degree 10 arithmetically Gorenstien curve that allows us to compute the map θ

  If p ∈ P 2 \ Z, then rank(N Z ⊗ C(p)) = 4 as (1) remains exact after tensorization by C(p), residue field of p. So p is mapped to the point in P 4 whose homogeneous coordinates solve the linear system[START_REF] Bertolini | Instability of projective reconstruction from 1-view near critical configurations in higher dimensions[END_REF] (N Z ⊗ C(p))

  b -Φ 13 a = 0, where a = (Ψ 1 ) 21 b 12 -(Ψ 1 ) 22 b 13 + (Ψ 1 ) 23 b 23 + (Ψ 1 ) 24 b 14 -(Ψ 1 ) 25 b 24 + (Ψ 1 ) 26 b 34 , b = (Ψ 1 ) 31 b 12 -(Ψ 1 ) 32 b 13 + (Ψ 1 ) 33 b 23 + (Ψ 1 ) 34 b 14 -(Ψ 1 ) 35 b 24 + (Ψ 1 ) 36 b 34 and (b 12 , . . . , b 34 ) are the Plücker coordinates of the line through the points (det((ψ 2 ) 1), det((ψ 2 ) 2), det((ψ 2 ) 3), det((ψ 2 ) 4))

- 1 :

 1 G(1, 3) → P 4 . Input: ideal I K of a general line congruence. Output: ideals I S and I C where S = K ∪ V, C = K ∩ V . ring r = 0, (x(0..5), a(1..268)),dp; ideal i k ;\\ This ideal has to be given as input. The first generator is the Klein hyperquadric. option(redSB); ideal k 1 = std(i k ); \\ reduced standard basis of i k \\ In the following lines, we compute general polynomials that will become a reduced standard basis of i s matrix mon3[START_REF] Arrondo | Sols On conguences of lines in projective space[END_REF][46] = x(5) 3 , x(4) * x(5) 2 , x(3) * x(5) 2 , x(2) * x(5) 2 , x(1) * x(5) 2 , x(0) * x(5) 2 , x(4) 2 * x(5), x(3) * x(4) * x(5), x(2) * x(4) * x(5), x(1) * x(4) * x(5), x(0) * x(4) * x(5), x(3) 2 * x(5), x(1) * x(3) * x(5), x(0) * x(3) * x(5), x(2) 2 * x(5), x(1) * x(2) * x(5), x(0) * x(2) * x(5), x(1) 2 * x(5), x(0) * x(1) * x(5), x(0) 2 * x(5), x(4) 3 , x(3) * x(4) 2 , x(2) * x(4) 2 , x(1) * x(4) 2 , x(0) * x(4) 2 , x(3) 2 * x(4), x(1) * x(3) * x(4), x(0) * x(3) * x(4), x(2) 2 * x(4), x(1) * x(2) * x(4), x(0) * x(2) * x(4), x(1) 2 * x(4), x(0) * x(1) * x(4), x(0) 2 * x(4), x(3) 3 , x(1) * x(3) 2 , x(0) * x(3) 2 , x(1) 2 * x(3), x(0) * x(1) * x(3), x(0) 2 * x(3), x(2) 3 , x(1) * x(2) 2 , x(0) * x(2) 2 , x(1) 2 * x(2), x(0) * x(1) * x(2), x(1) 3 ; matrix cp1[46][1] = a(1..46); poly p 1 = (mon3 * cp1)[1, 1] + x(0) 3 ; matrix cp2[46][1] = a(47..92); poly p 2 = (mon3 * cp2)[1, 1] + x(0) 2 * x(1); matrix cp3[46][1] = a(93..138); poly p 3 = (mon3 * cp3)[1, 1] + x(0) * x(1) 2 ; matrix cp4[45][1] = a(139..183); poly p 4 = (submat(mon3, intvec(1), intvec(1..45)) * cp4)[1, 1] + x(0) 2 * x(2); matrix mon4[1][85] = x(5) 4 , x(4) * x(5) 3 , x(3) * x(5) 3 , x(2) * x(5) 3 , x(1) * x(5) 3 , x(0) * x(5) 3 , x(4) 2 * x(5) 2 , x(3) * x(4) * x(5) 2 , x(2) * x(4) * x(5) 2 , x(1) * x(4) * x(5) 2 , x(0) * x(4) * x(5) 2 , x(3) 2 * x(5) 2 , x(1) * x(3) * x(5) 2 , x(0) * x(3) * x(5) 2 , x(2) 2 * x(5) 2 , x(1) * x(2) * x(5) 2 , x(0) * x(2) * x(5) 2 , x(1) 2 * x(5) 2 , x(0) * x(1) * x(5) 2 , x(0) 2 * x(5) 2 , x(4) 3 * x(5), x(3) * x(4) 2 * x(5), x(2) * x(4) 2 * x(5), x(1) * x(4) 2 * x(5), x(0) * x(4) 2 * x(5), x(3) 2 * x(4) * x(5), x(1) * x(3) * x(4) * x(5), x(0) * x(3) * x(4) * x(5), x(2) 2 * x(4) * x(5), x(1) * x(2) * x(4) * x(5), x(0) * x(2) * x(4) * x(5), x(1) 2 * x(4) * x(5), x(0) * x(1) * x(4) * x(5), x(0) 2 * x(4) * x(5), x(3) 3 * x(5), x(1) * x(3) 2 * x(5), x(0) * x(3) 2 * x(5), x(1) 2 * x(3) * x(5), x(0) * x(1) * x(3) * x(5), x(0) 2 * x(3) * x(5), x(2) 3 * x(5), x(1) * x(2) 2 * x(5), x(0) * x(2) 2 * x(5), x(1) 2 * x(2) * x(5), x(0) * x(1) * x(2) * x(5), x(1) 3 * x(5), x(4) 4 + a(231) * x(3) * x(4) 3 , x(2) * x(4) 3 , x(1) * x(4) 3 , x(0) * x(4) 3 , x(3) 2 * x(4) 2 , x(1) * x(3) * x(4) 2 , x(0) * x(3) * x(4) 2 , x(2) 2 * x(4) 2 , x(1) * x(2) * x(4) 2 , x(0) * x(2) * x(4) 2 , x(1) 2 * x(4) 2 , x(0) * x(1) * x(4) 2 , x(0) 2 * x(4) 2 , x(3) 3 * x(4), x(1) * x(3) 2 * x(4), x(0) * x(3) 2 * x(4), x(1) 2 * x(3) * x(4), x(0) * x(1) * x(3) * x(4), x(0) 2 * x(3) * x(4), x(2) 3 * x(4), x(1) * x(2) 2 * x(4), x(0) * x(2) 2 * x(4), x(1) 2 * x(2) * x(4), x(0) * x(1) * x(2) * x(4), x(1) 3 * x(4), x(3) 4 , x(1) * x(3) 3 , x(0) * x(3) 3 , x(1) 2 * x(3) 2 , x(0) * x(1) * x(3) 2 , x(0) 2 * x(3) 2 , x(1) 3 * x(3), x(2) 4 , x(1) * x(2) 3 , x(0) * x(2) 3 , x(1) 2 * x(2) 2 , x(0) * x(1) * x(2) 2 , x(1) 3 * x(2);

  22 ϕ 31 ϕ 32 ϕ 21 ϕ 23 ϕ 31 ϕ 33 ϕ 22 ϕ 23 ϕ 32 ϕ 33 ϕ 21 ϕ 24 ϕ 31 ϕ 34 ϕ 22 ϕ 24 ϕ 32 ϕ 34 ϕ 23 ϕ 24 ϕ 33 ϕ 34 ϕ 31 ϕ 32 ϕ 11 ϕ 12 ϕ 31 ϕ 33 ϕ 11 ϕ 13 ϕ 32 ϕ 33 ϕ 12 ϕ 13 ϕ 31 ϕ 34 ϕ 11 ϕ 14 ϕ 32 ϕ 34 ϕ 12 ϕ 14 ϕ 33 ϕ 34 ϕ 13 ϕ 14 ϕ 11 ϕ 12 ϕ 21 ϕ 22 ϕ 11 ϕ 13 ϕ 21 ϕ 23 ϕ 12 ϕ 13 ϕ 22 ϕ 23 ϕ 11 ϕ 14 ϕ 21 ϕ 24 ϕ 12 ϕ 14 ϕ 22 ϕ 24 ϕ 13 ϕ 14 ϕ 23 ϕ 24

  [START_REF] Bordiga | La superficie del 6 • ordine con 10 rette, nello spazio R 4 e le sue proiezioni nello spazio ordinario[END_REF] Φ 21 Φ 31 in g 1 . First, we verify that J is generated by g 1 , . . . , g 4 . In fact, it holdsThe equalities have to be checked by verifying that the coefficients of Φ 1i Φ 2j Φ 3h are the same on both sides. It is very long, but straightforward. As example, we check some of them.The first term we consider is Φ 13 Φ 23 Φ 31 in the g 5 relation. The coefficient on the left side is equal to det

	g 5 = -g 6 = -g 7 = -g 8 = -g 9 = -det(M ) = α 31 α 54 -α 34 α 51 α 34 α 31 α 64 -α 34 α 61 α 34 α 31 α 34 g 1 -α 32 α 34 g 2 + g 1 -g 1 -1 α 34 α 31 α 84 -α 34 α 81 α 34 g 1 -α 31 α 94 -α 34 α 91 α 34 g 1 -1 α 34 D c 2,3,4 g 1 -α 34 D α 32 α 54 -α 34 α 52 α 34 α 32 α 64 -α 34 α 62 α 34 g 3 -α 33 g 4 α 34 α 32 α 84 -α 34 α 82 α 34 α 32 α 94 -α 34 α 92 α 34 1 c 1,3,4 g 2 + 1 α 34 D g 2 + g 2 + g 2 + g 2 + c 1,2,4 g 3 . α 54 α 34 g 3 -α 64 α 34 g 3 -α 84 α 34 g 3 -α 94 α 34 g 3 -	α 33 α 54 -α 34 α 53 α 34 α 33 α 64 -α 34 α 63 α 34 α 33 α 84 -α 34 α 83 α 34 α 33 α 94 -α 34 α 93 α 34	g 4 g 4 g 4 g 4

  check the equality of the coefficients of Φ 11 Φ 21 Φ 31 in the det(M ) relation. On the left side, we have -c 147 . On the right side, we have

		α 54 α 34	α 31 det	α 32 α 33 α 62 α 63		-α 32 det		α 31 α 33 α 61 α 63	+ α 33 det	α 31 α 32 α 61 α 62	+
	+ α 51 det	α 32 α 33 α 62 α 63	-α 52 det	α 31 α 33 α 61 α 63	+ α 53 det	α 31 α 32 α 61 α 62	=
	=D	   -	α 54 α 34	det	 	α 31 α 32 α 33 α 31 α 32 α 33 α 61 α 62 α 63	  + det	 	α 51 α 52 α 53 α 31 α 32 α 33 α 61 α 62 α 63	 	  
	and so the equality holds.
	Now, we + 1 α 34 D	c 234 det

  54 Φ 22 + α 64 Φ 23 Φ 31 + α 84 Φ 32 + α 94 Φ 33

	α 34	Φ 11 -	α 31 α 34 α 51 α 54	Φ 22 -	α 31 α 34 α 61 α 64	Φ 23	-α 31 Φ 31 --α 31 α 34 α 31 α 34 α 81 α 84 α 91 α 94 Φ 33	Φ 32 -
	D α 34	Φ 12 -	α 32 α 34 α 52 α 54	Φ 22 -	α 32 α 34 α 62 α 64	Φ 23	-α 32 Φ 31 --α 32 α 34 α 32 α 34 α 82 α 84 α 92 α 94 Φ 33	Φ 32 -
	D α 34 α 0 Φ 13 -α 34 Φ 21 -α 63 α 64 α 33 α 34 α 33 α 34 Φ 23 α 53 α 54	Φ 22		-α 33 Φ 31 -	α 33 α 34 α 83 α 84	Φ 32 -
	D α 34	-α 94	α 32 α 34 α 52 α 54	+ α 54	α 32 α 34 α 92 α 94	=	D α 34	α 34	α 52 α 54 α 92 α 94

Now, we give an algorithm that summarizes the previous discussion, and that can be easily implemented in Singular [START_REF] Decker | Singular 4-1-0 -A computer algebra system for polynomial computations[END_REF], for example.

Input: projection matrices P 1 , P 2 , P 3 , Q 1 , Q 2 , Q 3 . Output: projection matrices ϕ 1 , ϕ 2 , ϕ 3 , ψ 1 , ψ 2 , ψ 3 .

(1.1) Given the 5 × 1 matrix V Y = [y 0 , . . . , y 4 ], compute P 1 • V Y, P 2 • V Y, P 3 • V Y and the 9 × 8 matrix M . (1.2) Compute the matrix N X = A -BD -1 C as in section 5, and the ideal I X of the maximal minors of N X . (2) Choose a general 3 × 2 matrix M L of numbers, compute M Γ = N X M L , and its maximal minors q 0 , . . . , q 5 in the above order. (3) Define the map θ by setting θ(x i ) = q i , i = 0, . . . , 5, and compute

to equation ( 3). (4.2) Compute g 0 , . . . , g 4 , maximal minors of N Z with alternating signs, and set I Z the ideal they generate. (4.3) Define the map π by setting π(y i ) = g i , i = 0, . . . , 4.

(5.1) Compute

, where (. . . ) j means the degree j part of the ideal in parenthesis. (5.2) Compute I Y = I K ∩ i=1,2,3 I α i . (6.1) Compute the third matrix in the minimal free resolution of I Y and take the submatrix with linear entries. (6.2) Put this matrix in the form of the proof of Proposition 6.1. (6.3) Compute matrices Φ i , Ψ j , 1 ≤ i, j ≤ 3, from the last matrix, and the associated projection matrices ϕ i , ψ j . Now, we describe the converse construction.

(1) Given the projections ϕ i , ψ j : P 3 → P 2 , we can compute the maps Φ i , Ψ j and the critical locus Y ⊂ G(1, 3) for the reconstruction problem for projections of lines, as explained in section 6. Moreover, we compute the line congruence K and the three α-planes that are the irreducible components of Y. (2) As explained in section 3, we compute a Veronese surface V ⊂ G(1, 3) such that K ∪ V is an arithmetically Gorenstein surface of degree 13 and K ∩ V is a degree 10 arithmetically Gorenstein curve. (3) Let θ -1 : R → T G be the map associated to K ∩ V , where

where X is a Bordiga surface. (4) From Theorem 5.1, we get projections P i , Q j :

is the associated critical locus for the reconstruction problem. As before, we give the corresponding algorithm. It can be implemented in Singular [START_REF] Decker | Singular 4-1-0 -A computer algebra system for polynomial computations[END_REF].

(1.1) Compute Φ i , Ψ j , 1 ≤ i, j ≤ 3, as in section 6.

(1.2) Compute the critical locus Y for the reconstruction problem for lines, and its irreducible components. (2) In the line congruence K contained in Y, compute a degree 10 arithmetically Gorenstein curve C = K ∩ V whose ideal is generated by 5 quadrics q 0 , . . . , q 4 in addition to the Klein hyperquadric x 0 x 5 -

\\ This ideal will contain the relations among the a(j)'s poly sp;

\\ the following are local variables int i; int j; int k; int f l; ideal mx = x(0..5); matrix mx3 [START_REF] Arrondo | Sols On conguences of lines in projective space[END_REF][56] = std(mx 3 ); \\ monomials of degree 3 matrix mx4 [START_REF] Arrondo | Sols On conguences of lines in projective space[END_REF][126] = std(mx 4 ); \\ monomials of degree 4 matrix lp1 [START_REF] Arrondo | Sols On conguences of lines in projective space[END_REF][8] = std(lead(k 1 ));

\\ leading monomials of k 1 for(i = 1; i <= 8;

\\ normalized reduced standard basis of k 1 ;

\\ we compute the normal form of p1 with respect to k 1 and we collect the coefficients of the monomials in the ideal rel sp = p1; for(j = 56; j >= 1;

\\ repeat verbatim the above computation for sp = p2, p3, p4 and we collect the coefficients of the monomials in the same ideal rel sp = p2;

. . . \\ we compute the normal form of p5 with respect to k 1 and we collect the coefficients of the monomials in the same ideal rel sp = p5; for(j = 126; j >= 1;

\\ we compute a standard basis of rel, and we reduce p1, . . . , p5 modulo it. After this step, they belong to i k rel = std(rel); poly p11 = reduce(p1, rel); [1, k]; }; }; }; \\ we choose randomly the values of a(182), a(183) and compute a standard basis of rel: in such a way, every coefficient has a numerical value that we substitute in p11, . . . , p51 ideal rel1 = a(182) -random(-5, 5), a(183) -random(-5, 5), rel; rel1 = std(rel1); poly p12 = reduce(p11, rel1); poly p22 = reduce(p21, rel1); poly p32 = reduce(p31, rel1); poly p42 = reduce(p41, rel1); poly p52 = reduce(p51, rel1); ideal i s = i k [START_REF] Arrondo | Sols On conguences of lines in projective space[END_REF], p12, p22, p32, p42, p52; ideal i v = quotient(i s , i k ); ideal i c = i v , i k ;