Towards the degree of the PGL(4)-orbit of a cubic surface

doi: 10.4418/2020.75.2.3

TOWARDS THE DEGREE OF THEPGL(4)-ORBIT OF A CUBIC SURFACE

ELISA CAZZADOR - BJØRN SKAULI

We study the action of the group PGL(4)on the parameter spaceP¹⁹ of complex cubic surfaces. Specifically, we look at how the techniques used by Aluffi and Faber in [1] can be extended to compute the degree of the orbit closureOof a general cubic surface. We study the base locus of the induced rational mapP¹⁵99KO⊂P¹⁹, and the first steps in resolving this rational map by successively blowing up the reduced base locus.

1. Introduction

A complex cubic surfaceSinP³is the vanishing locus of a homogenous degree- 3 form of the type

F(x) =a₀x³₀+a₁x²₀x₁+· · ·+a₁₉x³₃.

It is clear that cubic surfaces are parametrized byPSym³(C⁴)^∗'P¹⁹. However, two isomorphic surfaces correspond to different points inP¹⁹and the simplest way this can happen is when changing coordinates. A natural question would then be: Given a fixedS as above, which other cubic surfaces arise fromSby coordinate change? In other words, we are asking to describe the orbitOofS under the action of the group PGL(4)on the parameter spaceP¹⁹.

We would like to study the geometry of O: since this just forms a locally closed subset inP¹⁹, we will rather consider its closureO. A first step in this

Received on September 14, 2019

direction is to compute its degree. The latter will depend on the choice of the surfaceSand in this paper we will primarily focus on the case whereSis chosen to be general.

In the special case of the Cayley cubic, a surface with four distinct nodes, the degree of the orbit closure is already known. In [7], this number is computed to be 305, based on counting cubic surfaces with 4 distinct double points passing through 15 general points.

WhenSis general, the degree of the orbit closure will be significantly higher and different techniques will be needed. One could start by looking at the map φ: PGL(4)→P¹⁹, sending the class of a matrix to its pre-composition withF. The image of this map is the orbit Oand computing the degree of its closure would amount to count the number of points in the intersection of O with a general linear subspace of complementary dimension.

We can count the number of such points using intersection theory by finding a pair (V,e φ)e such that Ve is a compactification of PGL(4) andφe a dominant regular morphism fromVe to Oextending φ and the intersections of the pull- back of a hyperplane class byφeis transversal. Then we can simply compute φe^∗c₁(O

P¹⁹(1))¹⁵.

The first naïve compactification one could think of isPHom(C⁴,C⁴)'P¹⁵, which can be as well equipped with the pre-composition map which naturally extendsφ. Unfortunately, this pair is not good enough since the given map is not regular. From a computational viewpoint, issues come from the fact that the pull-back classes we are considering will intersect in positive dimension.

The strategy that we would like to pursue here is to find an explicit resolution ofφ where it is possible to keep track of how the intersections change. This approach was already considered by Aluffi and Faber who studied the case of plane curves of any degree. What we are going to do in the present paper is to adapt many of the ideas contained in there. In particular, we decide to regularize φ by a sequence of blow-ups at smooth centers. We will start by describing the support of the base locus Bs(φ)from a set-theoretical point of view. We will then study the first steps towards the resolution ofφby successively blowing up the reduced components of the base locus.

Four of these steps will be analyzed, though currently it is not clear if they will be sufficient to give the desired resolution. This difficulty reflects an im- portant difference from the case of plane curves: here the base locus of φ has many components, and this is a consequence of the fact that a general cubic surface contains 27 distinct lines. More specifically, we will see that problems can possibly arise from those morphisms in P¹⁵whose image is spanned by a point contained in one of these lines.

The aim of this paper is to present a report of an on-going project, where the

remaining work that needs to be done regards not only proving or disproving the existence of further components to blow up. Indeed, as mentioned above, there is also a computational aspect, namely showing how the different steps in the resolution contribute in finding the degree of O. These computations will not be analyzed here, since the results would currently be very partial. They are hopefully going to appear in a future paper, as the natural conclusion of the work illustrated here. For this second part as well, we believe that a considerable inspiration could be taken by the techniques developed in [1].

An alternative approach to the same problem has been recently explored by Brustenga i Moncusí, Timme and Weinstein in [3]. There is however a signifi- cant difference between the methods. Indeed, in their paper the computation of the degree of the orbit closure is treated from a more numerical perspective. The idea is to count the number of solutions of a system of polynomial equations in an affine variety using homotopy continuation and monodromy methods. As a result, for a generalS, this number turns out to be 96120. On the other hand, applying intersection theory in the context of resolutions of singularities gives a more geometric flavor and we believe that this will help to shed some light on a complete understanding on the studied phenomena.

The problem was firstly introduced to us from the 27 Questions on Cubic Surfaces(see [6]), in view of the First Meeting on Cubic Surfaces, that was held in Oslo on May 13, 2019. We would like to thank: Kristian Ranestad and Corey Harris for the valuable discussions and the patience with the many questions, Paolo Aluffi for very nice explanations about his paper [1], Maddie Weinstein for stimulating conversations, the anonymous referees for all the corrections and suggestions.

2. Setup

In this section we will first describe the action of PGL(4)on the parameter space of cubic surfaces. This will naturally produce rational maps

P¹⁵'PHom(W,W)99KPSym³(W^∗)'P¹⁹,

one for every fixed cubic. If the latter is chosen to be general, it will be possible to illustrate how this map can be used to compute the degree of the orbit closure.

Throughout the paper we will work over the fieldCof complex numbers.

2.1. The action ofPGL(4)

Let us denote with W the 4-dimensional vector space C⁴. A complex cubic surfaceS ⊂PW is the zero set of a homogeneous degree-3 polynomial in four

variables, say

F(x) =a₀x₀³+a₁x²₀x₁+· · ·+a₁₈x₂x²₃+a₁₉x₃³,

which corresponds to a point [F]:= [a₀:a₁:· · ·:a₁₉]in the parameter space F:=PSym³(W^∗). The group PGL(4)acts onFby pre-composition (or, equiv- alently, by coordinate change):

PGL(4)× F → F

(α,[F(x)])7→[F(α(x))]

For a fixedS(hence for a fixedF), this yields a mapφ: PGL(4)→ F, whose image is by definition the orbit OofF. Moreover, the fiberφ⁻¹(F)is the set of automorphisms of W leaving F unchanged, so it corresponds to group of linear automorphisms ofS. Our object of study is the degree ofOinF: to this purpose, we first need to understand the dimension ofOand the degree ofφ.

Let us denote with V the spacePHom(W,W) of nonzero endomorphisms ofW up to projective equivalence, which is also canonically isomorphic to the space of matricesP(W^∗⊗W).

Lemma 2.1. LetSbe a cubic surface with finite group of linear automorphisms.

ThendimO=15.

Proof. By hypothesisφ is a finite map, so dimO=dim PGL(4). But dimO= dimOand PGL(4)embeds as an open subsetV, whose dimension is 15.

From now on we will considerSto begeneral, meaning that its correspond- ing point inFlies in some proper Zariski open subset.

Lemma 2.2. IfSis a general cubic surface, the above mapφ has degree1.

Proof. We will prove that each fiber ofφ consists of a single point. Suppose that there exist two pointsα1,α2of PGL(4)with the property thatF(α1(x)) = F(α₂(x)): then the compositeα₁⁻¹◦α₂ would be a linear automorphism ofS.

But a general cubic surface has no nontrivial linear automorphisms (see [5]), so α1=α2.

2.2. How to compute the degree ofO

As we noticed in the proof of Lemma 2.1, we can see PGL(4)as an open subset of V; in particular the map φ can be understood as a rational map V 99KF, which, by abuse of notation, we will keep callingφ. The strategy from [1] that we want to apply here is to resolveφby a sequence of blow-ups inVand finally get a regular mapφ:e V → Fe , whereVeis a smooth compactification of PGL(4)

and imφe=O. The blow-ups will also produce a morphismπ:V → V, such thate the following diagram commutes:

PGL(4) Ve F

PGL(4) V F

φe

π φ

Given this construction, we can compute the degree d of O as follows:

letφe∗: CH(Ve)→CH(F) be the push-forward map between the corresponding Chow rings and let us recall thatOis a 15-dimensional subvariety ofF. Then by definitiond=^R_F[O]·H¹⁵, where^R_F(·)denotes the degree of the 0-dimensional part, whileHdenotes the hyperplane class in CH(F)'Z[H]/H¹⁶. On the other hand, by constructionVe dominatesO, so degφe·[O] =φe∗(1). Then, using the projection formula, we find:

degφe·d= Z

F

φe∗(1·φe^∗H¹⁵) = Z

Ve

φe^∗H¹⁵. (1) Definition 2.3. With notation as above, we define the predegree of O to be R

Ve(φe^∗(H))¹⁵.

Note that, even whenShas nontrivial linear automorphisms, it is possible to use equation (1) to find the degree ofOby dividing the predegree by the order of the group of linear automorphisms. In the general case we have the following result:

Proposition 2.4. For a general cubic surface S, the degree of O equals its predegree.

Proof. Indeed, thanks to Lemma 2.2, we know that ifS is general, degφe=1;

then the expression (1) gives the desired equality.

The first step towards the resolution of φ is to understand its base locus Bs(φ). To this purpose we note that the linear system definingφ is spanned by a certain set of hypersurfaces having a nice geometric interpretation.

Definition 2.5. LetS=V(F)be a cubic surface inPW. For every p∈PW, the point condition Ppis defined as:

Pp={α∈ V |F(α(p)) =0}

i.e. the zero locus ofF(α(p))as a polynomial inα.

Since the point conditions span the linear system definingφ, the base locus Bs(φ)can be identified with the intersection ^T_p∈PWPp. After blowing up this locus inV, we will get a new rational map, whose base locus will be described by the intersection of the proper transforms of the point conditions, and so on.

Moreover, if we denote by Pep the proper transform of Pp in V, we see thate d=^R

Ve[Pe_p]¹⁵.

Although the main focus of this paper is to illustrate the several steps needed to resolveφ, we would like to mention here a very important proposition, which can be (repeatedly) used to tell how the various blow-ups contribute in the com- putation of the degree ofO.

Proposition 2.6 ([1, Proposition 3.2]). Let i:B→V be an inclusion of non- singular projective varieties, and let X ⊂V be a codimension-1 subvariety, smooth along B. LetV be the blow-up of V along B, and lete X be the propere transform of X . Then

Z

Ve

[X]e^dimV = Z

V

[X]^dimV− Z

B

([B] +i^∗[X])^dimV c(N_B/V) ,

where c(N_B/V)denotes the total Chern class of the normal bundle of B in V . In our situation, the role of V andX will be played by V and P_p, while Bwill represent each time a component of the reduced base locus that we are blowing up. Since the point-conditions are cubic hypersurfaces inV, we have R

V[Pp]¹⁵=3¹⁵. Then the degree ofOwill be 3¹⁵−n₁− · · · −n_k, where then_i’s are the contributions of the blown up loci that can be explicitly computed using Proposition 2.6.

At each step, the most difficult part will be to computec(NB/V)in the Chow ring CH(B). This motivates us to look for a resolution, by picking a suitable sequence of blow-ups that allows to handle this computation easily.

The contributions coming from the sequence of blow-ups is left for a future paper, that is thought to be the natural continuation of the present one.

3. Towards the resolution ofφ

In this section, we will describe the first steps necessary to regularizeφ accord- ing to the strategy described in 2.2. It is not yet clear if these are enough or if more blow-ups are required. An important difference from the case of plane curves studied in [1] is that the base locus Bs(φ) has not only one, but many components, reflecting the fact that a general cubic surface contains 27 distinct lines.

3.1. The base locus ofφ

With the next proposition we are going to describe Bs(φ) as a set. To this purpose, we look atV as the space of matricesP(W^∗⊗W), together with the Segre embedding

PW^∗×PW,→P(W^∗⊗W) given by

([k0:· · ·:k₃],[q0:· · ·:q₃])7→









k₀q₀ k₁q₀ k₂q₀ k₃q₀ k₀q₁ k₁q₁ k₂q₁ k₃q₁ k₀q₂ k₁q₂ k₂q₂ k₃q₂ k₀q₃ k₁q₃ k₂q₃ k₃q₃







 ,

wherek^⊥:={x∈PW|k₀x₀+· · ·+k₃x₃=0}is the kernel of such a matrix and q:= [q0:· · ·:q₃]its image.

Proposition 3.1. LetS=V(F)be a general smooth cubic surface inPW . Let φbe the map defined above. ThenBs(φ)is supported at the union of two closed components B and C, with:

(i) B'PW^∗× S;

(ii) C' ∪²⁷_i=1Ci,

where the C_i’s are the irreducible components of C and each C_iis isomorphic to P⁷.

Proof. The mapφis not defined over the set

{α ∈ V |F(α(x))≡0}={α∈ V |imα⊂V(F)}.

Since S is taken to be general, its linear subspaces are points inS and the 27 lines, that we denote by`1, . . . , `27. We can write the base locus as

B∪C, where

B:={α∈ V |rkα=1,imα∈ S}, C:={α ∈ V |rkα≤2,imα⊆`ifor somei}.

(i) The matrices inBare parametrized by the choice of a point inS and the choice of a 4-tuple of coefficients inPW^∗(indeed each column must be a multiple of the chosen point). HenceB'PW^∗× S.

(ii) RegardingC, it consists of 27 components{C_i}²⁷_i=1, whereCiis the space of matrices whose image is spanned by`i. So for everyiwe can make the identificationC<sub>i</sub>'PHom(W,U), whereUis the 2-dimensional subspace ofW for whichP(U) =`i. This is a 7-dimensional projective linear space and in particular we getC' ∪²⁷_i=1P⁷.

Remark 3.2. Alternatively, one can see the the elements of a fixedC_i as the sum of two rank-1 matrices parametrized by the choice of a point on the given line and the choice of a 4-tuple of coefficients inPW^∗. In other words,Ci is the union of the span of all pairs of points in PW^∗×`i (including the degenerate case in which the two points coincide), so we are describing the secant variety Sec2(PW<sup>∗</sup>×`i)'Sec2(P³×P¹), which is aP⁷.

Remark 3.3. The subset, PGL(4)⊂ V does not intersect Bs(φ), so as we re- solve the rational mapφ, we still get compactifications of PGL(4).

Remark 3.4. The above proof actually says more: the two componentsBand Cintersect in

B∩C={α ∈ V |rkα=1,imα is a point on`ifor somei}.

In particular, this implies the following Corollary.

Corollary 3.5. Let Ci,i=1, . . . ,27be the components of C, each isomorphic to P⁷. Then

B∩C_i'PW^∗×`i. Moreover, for i6=j we have

C_i∩C_j' (

PW^∗ if`i∩`j6=/0 /0 otherwise

As we have mentioned at the end of Section 2, since Bs(φ)has many com- ponents, there are many ways of resolving the map. The following order of blow-ups at smooth centers is suited for relating the base loci of the induced maps to properties of point conditions inV.

We start by blowing upValong the componentB'PW^∗× S: this produces a morphismπ₁:V₁→ V and an exceptional divisorE₁⊂ V₁. After blowing up B, the proper transforms of the point condition, denoted byPp⁽¹⁾, will define a new rational mapφ₁:V₁99KF. Note thatB∩PGL(4) = /0 inV, soV₁contains an open dense subset isomorphic to PGL(4)and with a little abuse of notation we will indicate it using the same symbol. Let us denote withC_i⁽¹⁾the proper transform ofC_iinV₁for everyi.

Claim 3.6. The base locus Bs(φ₁) is supported on the 27 componentsC_i⁽¹⁾’s, which are disjoint, plus a further component, denoted byB₁, contained in the exceptional divisorE₁, intersecting theC⁽¹⁾_i ’s.

We will chooseB₁ to be the center of the second blow-up. As before, this will produce a new morphismπ₂:V₂→ V₁, together with an exceptional divisor E₂⊂ V₂. Again, the proper transforms of the point conditions, denoted byPp⁽²⁾, will define a rational mapφ₂:V₂99KF.

Claim 3.7. The support of Bs(φ₂)contains the 27 pairwise disjoint proper trans- formsC_i⁽²⁾’s and a subvariety, denoted byB₂, which has a dominant 2: 1 map to B.

Note that it is not clear whether the subvarietyB₂is irreducible or not. What we will prove is that it must consist of either 1 or 2 components. Moreover, we need to observe that Claim 3.7 refers to an inclusion, but not an equality, so there might be some other components in Bs(φ₂), namely the ones dominating the intersectionsB∩Ci'PW^∗×`i.

If we assume that we have exactly the components listed in above, we can proceed by blowing upB₂. We get as usual a mapπ₃:V₃→ V₂, an exceptional divisorE₃⊂ V₃ and a rational mapφ3:V₃99KV induced by the proper trans- forms of point conditions. We expect no component of the base locus ofφ3to dominateB. In fact, one might hope that the only components of Bs(φ₃)are the C⁽³⁾_i , and that blowing up these components resolves the rational map.

We summarize the construction in Figure 1, which also fixes notation for the rest of the section.

3.2. The base locus after blowing upB

We now aim to prove Claim 3.6; in particular, we are interested in giving the set-theoretical description ofB₁:=E₁∩Bs(φ₁).

To this purpose, we recall thatBis embedded inVvia the Segre embedding.

In particular, for everyα∈ V, we may identify the spaceT_αV with the quotient (W^∗⊗W)/αC. Letα = (k,q)be a point inBand let us denote withσ=TqS the tangent space ofSat the pointq.

Lemma 3.8. With the identification T_αV '(W^∗⊗W)/αC, we have:

(i) T_αB={τ∈W^∗⊗W|imτ⊂σ,τ(k^⊥)⊂q}/αC.

(ii) T_α(PW^∗×`i) ={τ∈W<sup>∗</sup>⊗W |imτ⊂`i,τ(k^⊥)⊂q}/αC.

G₁, . . . ,G₂₇ E₃⁽¹⁾ E₂⁽²⁾ E₁⁽³⁾ V₄ F

C₁⁽³⁾, . . . ,C₂₇⁽³⁾ E₃ E₂⁽¹⁾ E₁⁽²⁾ V₃ F

C₁⁽²⁾, . . . ,C₂₇⁽²⁾ B₂ E₂ E₁⁽¹⁾ V₂ F

C₁⁽¹⁾, . . . ,C₂₇⁽¹⁾ B₁ E₁ V₁ F

C₁, . . . ,C27 B V F

π4

φ4

π3

φ3

π2

φ2

π1

φ1

φ

Figure 1: The sequence of blow-ups (iii) The point condition P_pis non-singular atα and

T_αP_p={τ∈W^∗⊗W|τ(p)⊂σ}/αC.

Proof. The ideas in this proof are essentially the same of [1, Lemma 2.1].

(i) The (5-dimensional) tangent space ofBatαis T_αB=T_k(PW^∗× {q})⊕T_q({k} × S)

={k⁰⊗q∈W^∗⊗W |k⁰∈PW^∗}

kC ⊕{k⊗q⁰∈W^∗⊗W |q⁰∈σ} qC

={τ∈W^∗⊗W|imτ =q}

kC ⊕{τ∈W^∗⊗W|kerτ=k^⊥, imτ⊂σ}

qC .

The two spaces in the direct sum decomposition are both contained in the space

{τ∈W^∗⊗W|imτ ⊂σ,τ(k^⊥)⊂q}

(k⊗q)C ,

which is also of dimension 5, so they coincide.

(ii) Similarly we obtain the description forT_α(PW^∗×`i).

(iii) A line passing throughαcan be written asγ_α(s) =α+τs, for someτ∈ V.

Note that since imα =q∈ S, then F(γ_α(0)(p)) =F(α(p)) =0. The intersection multiplicitym_α(Pp·γ)is by definition the order of vanishing

ordt=0[F((α+τs)(p))],

so the lineγ is tangent toP_p if and only if that order is greater or equal than 2. By taking the Taylor expansion we get

F((α+τs)(p)) =F(α(p)) +

3

∑

i=0

∂F

∂xi

α(p)

τi(p)s+. . . ,

whereτi(p)denotes thei-th coordinate ofτ(p). Hence we need the con- stant and the linear term of this expression to vanish. We already know thatF(α(p)) =0, while∑i

∂F

∂x_i

qτi(p) =0 if and only ifτ(p)⊂σ, that is exactly the condition we claimed. The above computation says more:

ifτ(p)6⊂σ, then the lineα+τsintersectsPpwith multiplicity 1 atα, so Ppis non-singular atα.

We will also need a similar lemma describing various tangent spaces at points ofCi.

Lemma 3.9. For every pointα∈C_i, we have:

(i) T_αCi={τ∈W^∗⊗W|imτ⊂`i}/αC. (ii) T_αP_p={τ∈W^∗⊗W|τ(p)⊂T_α(p)S}/αC.

Proof. (i) Since eachCi'P⁷is embedded inVas a linear space, if we iden- tify theC_iwith nonzero matrices with image in`i, then the tangent space to this linear space at any point is simply the linear space itself, i.e. the matrices with image in`i.

(ii) Exactly as the proof of Lemma 3.8(iii)

Lemma 3.10. After blowing up B₁, the C_i⁽¹⁾are all disjoint inV₁.

Proof. Recall thatE₁ is defined asP(NB/V), with N_B/V 'TV/T B. Then the intersectionC_i⁽¹⁾∩E₁is the projectivization of the image ofTCivia the compo- sition

TC_i,→TV →TV/T B.

We need to prove that ifCiandCjintersect inV, thenC_i⁽¹⁾andC⁽¹⁾_j are disjoint in the blow-upV₁. We can check this fiberwise and show that for everyα∈Ci∩Cj, the images ofT_αC_iandT_αC_jinT_αV/T_αBdo not intersect.

First observe that blowing upValongBaffectsC_ias if it was blown up along PW^∗×`i, producing an exceptional divisorF_i :=P

TC_i T(PW^∗×`_i)

, embedded in E₁. We may therefore instead prove that the image ofC⁽¹⁾_j inFiis the empty set, i.e. thatT_αC_i∩ hT_αC_j,T_α(PW^∗×`i)iis contained inT<sub>α</sub>(PW<sup>∗</sup>×`i). Writeqfor the intersection`i∩`j andσ forTqS. Knowing the description of the tangent spaces in Lemma 3.8 and Lemma 3.9 and recalling that the two lines`i, `j span σ, we obtain

hT_αC_j,T_α(PW^∗×`i)i={τ ∈W<sup>∗</sup>⊗W|imτ⊂σ,τ(k<sup>⊥</sup>)⊂`j}/αC. Intersecting this span withT_αC_iwe obtain exactly the tangent spaceT_α(PW^∗×

`i), soC_i⁽¹⁾andC⁽¹⁾_j are disjoint in the blow-up.

The tangent spaces appearing in Lemma 3.8 are also essential to describe the base locus ofφ₁.

Proposition 3.11. The base locusBs(φ₁) of the rational map φ₁:V₁99KF is supported on

B₁∪ {C⁽¹⁾₁ , . . . ,C₂₇⁽¹⁾},

where B₁ is aP⁵-subbundle of E₁. Moreover, B₁= (^T_p∈PWPp⁽¹⁾)∩E₁ both set and scheme-theoretically.

Proof. This result refers to [1, Proposition 2.2]. As observed earlier, the base locus ofφ₁is set-theoretically^T_pPp⁽¹⁾. In particular, a pointα₁∈E₁lying in the fiber ofα∈Bis also in Bs(φ1)if it is determined by a vector in^T_pT_αPpwhich is normal to B. Thanks to Lemma 3.8(iii), we see that the intersection of all tangent spaces to the point conditions atαis given by the 11-dimensional space Σα:={τ∈W^∗⊗W|imτ⊂σ}/αC. This containsT_αB(see again Lemma 3.8) and the quotientΣ_α/T_αBis a 6-dimensional subspace of the fiber ofN_B/V over α. Movingα, we get a rank-6 subbundle ofN_B/V, so aP⁵-subbundle ofE₁= P(N_B/V), as we wanted. TheC_i⁽¹⁾’s are also base loci, since the corresponding Ci’s were so.

The second statement can be proved fiberwise: indeed, the fiber of B₁, a linear subspace, is cut out by fibers of the various Pp⁽¹⁾∩E₁, which are linear spaces themselves.

Corollary 3.12. The component B₁ can be globally described as P ^T

pT_αP_p T B

and its intersection with C_i⁽¹⁾is the bundle overPW×`igiven by:

C_i⁽¹⁾∩B₁=P

TCi

T(PW^∗×`i)

.

Proof. The global description ofB₁ is straightforward from Proposition 3.11.

Regarding the intersectionC_i⁽¹⁾∩B₁, this coincides withC_i⁽¹⁾∩E₁. The descrip- tion then holds by the same arguments used for Lemma 3.10.

3.3. The base locus after blowing upB₁

We now address Claim 3.7: although a complete description of the components of Bs(φ₂)will not be given, we will show which of those components are the ones dominatingB'PW^∗× S.

So, let us denote withB₂ the closed subvariety of Bs(φ₂)∩E₂ dominating B. In order to understandB₂we will need to look at the intersection ofS with its tangent planes. We will focus on the points ofS lying in the subsetS₀:=

S \^T27_i=1`i. Note that for everyq∈ S<sub>0</sub>, the plane cubic curveTqS ∩ Sis either a node or a cuspidal curve and if`is a line in the tangent cone of such a cubic at its singular pointq, then the intersection multiplicity ism_q(`·(TqS ∩ S)) =3.

Definition 3.13. A line of matricesα+τsinV, withα= (k,q)∈Bis called a special lineifq∈ S₀,τ(k^⊥)6⊂qand the image ofτis contained in a line tangent to the cubic curveTqS ∩ S atq.

We would like to translate properties of points inB₂ to properties of points inBand as we will soon see it will be useful to observe the following:

Lemma 3.14. The base locusBs(φ₂)is disjoint from E₁⁽¹⁾.

Proof. This is just a rephrase of the second part of Proposition 3.11, thanks to which we know that the point conditionsPp⁽¹⁾intersectE₁transversely inV₁. Proposition 3.15. Letα₂ be a point of E₂and let us denote withα= (k,q)its image in B via the composite mapπ₁◦π₂. Suppose also that q∈ S₀. Thenα₂ is in B₂if and only if it can be written as the intersection of E₂with the proper transform inV₂ of a special line inV. Moreover, the set of suchα₂ is dense in B₂.

Proof. By definition, α₂∈Bs(φ₂) if and only if it is contained in the proper transform of a general point conditionPp⁽²⁾. In particular, ifα₂ is inB₂, it must represent a direction normal toB₁and tangent to a general point conditionPp⁽¹⁾

atα₁:=π₂(α₂). We can identify this direction with a smooth curve germγ_α1 aroundα1inV₁, satisfying normality toB₁and the tangency condition:

m_α1(γα1·Pp⁽¹⁾)≥2, for a generalp∈PW.

Note that, using the above identification, we can writeα2=E₂∩(γα1)⁽¹⁾. Thanks to Lemma 3.14 we can rephrase everything in terms of curve germs inV: indeed,γ_α1 turns out to be not only normal toB₁, but to the whole ofE₁, so we can think of it as the proper transform of a lineγα=α+τs⊂V, which is normal toBand intersects a general point conditionP_pwith multiplicity greater or equal than 3.

Denoting as usual with σ the tangent planeTqS, we can equivalently say that:

α₂∈B₂ ⇐⇒ α=E₂∩(α+τt)⁽²⁾,

with imτ ⊂σ andτ(k^⊥)6⊂q (see Lemma 3.8), such that for a general p we havem_α((α+τs)·Pp)≥3.

This description reduces to study a special class of lines throughαinV=V:

we divide in 3 cases, depending on the rank ofτ, that can be either 1,2 or 3.

If rkτ =3, then imτ =σ. In particular, for a general p∈PW, we have τ(p) =qp, whereqpis a point varying onσand different fromq. Then the span

hα(p) =q,τ(p) =qpi

is a general lineλpinσ passing throughqand(α+τs)(p)is a parametrization of such a line. Then for a general pwe have

m_α((α+τs)·Pp) =ordt=0(F((α+τs)(p)))

=mq(λp· S)

=mq(λp·(S ∩σ)) =2<3, so in this caseα₂is not in the base locus.

If rkτ=2, then imτ=`, where`is a line in the tangent planeσ. Again, for a generalp∈PW, the span ofα(p)andτ(p)is a line throughqand we are interested in computingmq(λp·(S ∩σ)). There are two possibilities: ifq6∈`, then for every two distinct points p₁ and p₂ inPW the linesλp₁ andλp₂ are distinct. In particular, for a generalp, the above multiplicity will be 2, so in this case as well,α₂is not in the base locus.

On the other hand, ifq∈`, then for a general pwe constantly haveλp=` andα₂ is in the base locus precisely whenm_q(`·(S ∩σ)) =3, i.e. when `is one of the two tangent lines at the node q (or the double tangent line in the degenerate case). Note the multiplicity computation makes sense since we are assuming thatq∈ S₀.

Finally, if rkτ =1, then imτ=q⁰, a point inσ different fromq(otherwise this would contradictτ(k^⊥)6⊂q). Then, arguing as above, for a general p, the span ofα(p) =qandτ(p) =q⁰is a constant line`andα2is in the base locus if and only ifmq(`·(S ∩σ)) =3. Note that the rank-1 matrices τ satisfying this property come from taking the closure of the space of rank-2 matrices described at the previous step.

The density statement is a consequence of the fact that S₀ is dense in S, since a component dominatesBif and only if it dominatesPW^∗× S₀⊂B.

Our knowledge about the components of the base locus of φ₂ can be sum- marized in the following:

Proposition 3.16. The components of the support ofBs(φ₂) that dominate a component of the original base locus Bs(φ) are C₁⁽²⁾, . . . ,C₂₇⁽²⁾ and the irre- ducible components of B₂. Moreover, the map (π₁◦π₂)|_B2 is a double cover of B, i.e. B₂consists of at most2irreducible components.

Proof. We just need to observe that B₂ is obtained by taking the closure of a subset ofE₂ whose fibers overBcorrespond to two special lines ofV (counted with multiplicity).

Remark 3.17. While theC⁽²⁾_i ’s are clearly irreducible, we are still left lo under- stand if alsoB₂is.

3.4. The base locus after blowing up theC_i⁽³⁾

The last part of the paper is devoted to proving the following result:

Proposition 3.18. After blowing up one of the components C_i⁽³⁾, corresponding to matrices with image contained in a line, there will be no remaining base locus over the points in Cicorresponding to matrices of rank2.

Since, up to this point, the centers of all blow-ups have been away from matrices of rank 2, we will for simplicity consider the base locus after blowing upC_iinV instead ofC_i⁽³⁾inV₂.

We now wish to study the intersection of the tangent spaces of all point conditions. To this end, we will study the image of matrices contained in the intersection of all the tangent spaces. From Lemma 3.9 (ii) we see that for every α∈Ci, the intersection of all the tangent spacesT_αPpis:

\

p∈W

{τ ∈W^∗⊗W|τ(p)⊂T_α(p)S}/αC. (2)

In fact, as we will prove now, this condition will imply that the image of the matrixτ must be contained in`i.

The proof relies on pencils of hyperplanes. The hyperplanes inWcontaining

`iare parametrized byH 'P<sup>1</sup>. A pencil of hyperplanes containing`i will be a morphismP¹→ H, and the degree of the pencil is the degree of this morphism (if it is nonconstant).

In this and the following lemma, we will work with the affine space W instead ofPW.

Lemma 3.19. Let α ∈Ci be a point corresponding to a rank-2 matrix with image`i, and letτ∈W^∗⊗W be such that the image ofτin T_αV 'W^∗⊗W/αC is in∩_pT_αPp. Then for any two-dimensional subspace U⊂W such thatα(U) =

`i, we haveτ(U)⊆`i.

Proof. From the two-dimensional subspaceU we can construct a degree-two pencilP₁of hyperplanes inW containing `i by assigning tou∈U the hyper- plane defined by the equation

3

∑

i=0

∂F

∂xi

α(u)

=0,

whereFis the general degree three polynomial defining the cubic surfaceS. We think ofP₁as assigning tou∈U the tangent plane ofSatα(s). This defines a map fromP(U)'P¹toH. This pencil will have degree two, as it is defined by degree two polynomials.

Assume for contradiction thatτ(U)6⊆`i. There are three cases:τ(U)is ei- ther a one-dimensional space not contained in`i, a two-dimensional space with one-dimensional intersection with `i, or a two-dimensional space with zero- dimensional intersection with`i. In all cases, we construct a second pencilP₂ of hyperplanes containing`i, by assigning tou∈U the hyperplane spanned by τ(u)and`i. This defines a mapP(U)99KHwhich is a priori at least rational, but extends to a morphismP(U)→ Hsince the domain is a curve.

The condition (2) states that P₁ andP₂ are equal. Indeed, condition (2) requires thatP₂(p) =hp, `iiis mapped toT_α(p)SinT_α(p)V. But this can only happen ifP₂(p) =P₁(p). However, this cannot be true in any of the three cases, as we will see in the following:

• Ifτ(U)is a one-dimensional space not contained in`i, thenP₂is constant, and therefore not equal toP₁.

• In the case whereτ(U) is two-dimensional and intersects`i in the one- dimensional spaceqC, P<sub>2</sub>(p)will be the hyperplane spanned by `i and τ(U)for anyp, so the pencil is constant.

• Ifτ(U)is a two-dimensional space intersecting`ionly in 0, thenP₂is a pencil of degree 1. Therefore, again, it cannot be equal toP₁.

From this lemma we can deduce that in fact the image ofτmust be in`i. Lemma 3.20. With notation as above, let α be a matrix of rank 2 in Ci. If τ∈^T_pT_αP_p, thenτis in the tangent space T_αC_i.

Proof. Letτ⁰be any element of∈W^∗⊗W that is mapped toτ. For any vector u∈W\kerα, it is possible to find a 2-dimensional subspaceU containing u such that α(U) =`i. Then, thanks to Lemma 3.19, we have τ<sup>0</sup>(u)∈`i. But sinceuwas arbitrarily chosen inW\kerα and this latter set spansW, we must have imτ⁰⊂`i.

Putting all this together we find that after blowing up a component of the base locus corresponding to matrices with image in a certain line, the remaining base locus is supported in the fibers over the rank-1 matrices.

Proposition 3.21. Let

PGL(4) V⁰ F

PGL(4) V F

φ⁰

π φ

be the the diagram associated to the blow-up ofV, along one of the components Ci 'P⁷ and let φ⁰:V⁰ 99KF be the induced rational map. If we denote by G_i the exceptional divisor over C_i and by Bs(φ⁰) the base locus of φ⁰, then π(Bs(φ⁰)∩G_i)is contained in thePW^∗×`i⊂Ciconsisting of rank-1matrices.

Proof. We will prove the statement fiberwise. Letα ∈Ci be a rank-2 matrix.

Then we must show that Bs(φ⁰)∩π⁻¹(α) is empty. The fiber π⁻¹(α) is the projectivization of (N_Ci/V)_α, the fiber of the normal bundle ofCi atα. If we denote with Pp the stict transform of a point condition, then Pp∩π⁻¹(α) is the projectivization of the quotientT_αP_p/T_αC_i. Therefore Bs(φ⁰)∩π⁻¹(α), is obtained by projectivizing ^T_pT_αPp/T_αCi. But by Lemma 3.20 we know that T

pT_αPpis actually contained inT_αCi, so the quotient described above must be trivial. After projectivizing, we see that Bs(φ⁰)∩π⁻¹(α)must be empty.

Remark 3.22. In our resolution ofφ:V 99KF, we actually want to blow up the proper transformsC_i⁽³⁾of theC_i inV₂. However, over the matrices of rank 2, the blow-downV₂→ V is an isomorphism. We can therefore conclude from Propostion 3.21 that also in this case there is no further base locus over the rank-2 matrices.

Having Proposition 3.21 been proved, the natural question to ask is:

Question 3.23. After blowing up theC_i⁽³⁾’s, is there any base locus over the subset of points that projects down to the locus of rank-1 matrices?

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ELISA CAZZADOR Department of Mathematics University of Oslo e-mail: [email protected] BJØRN SKAULI Department of Mathematics University of Oslo e-mail:[email protected]