DEGENERATIONS OF ( 1 , 7 ) -POLARIZED ABELIAN SURFACES
F. MELLIEZ and K. RANESTAD∗
Abstract
The moduli space of(1,7)-polarized abelian surfaces with a level structure was shown by Mano- lache and Schreyer to be rational with compactification the variety of powersum presentations of the Klein quartic curve. In this paper the possible degenerations of the abelian surfaces corresponding to degenerations of powersum presentations are classified.
1. Introduction
The moduli space A(1,7) of (1,7)-polarized abelian surfaces with a level structure was shown by Manolache and Schreyer to be rational with compac- tificationV (K4)a Fano 3-foldV22 [13]. Gross and Popescu obtain the same compactification ofA(1,7)with a different approach [10], but in neither case is the boundary V (K4)\A(1,7) discussed. The purpose of this paper is to describe this boundary. We show that every point onV (K4)correspond to a surface inP6invariant under the action of a groupG7, and we give a precise description of these surfaces.
More precisely, the(1,7)-polarized abelian surfaceAwith its level structure is embedded in P6 = PV0, where V0 is the Schrödinger representation of the Heisenberg group H7 of level 7. The embedding is invariant under the action ofG7, an extension ofH7 by an involution. The fixed points of this involution and its conjugates in G7 form an H7-orbit of planes P+2 and 3- spacesP−3. The 3-foldV (K4)parameterizes, what we denote by generalized G7-embedded abelian surfaces (cf. 2.9). Every such surface intersectsP+2 in a finite subscheme of length six, which we may classify by its type, namely the length of its components.
In this paper we prove the
Theorem 1.1. Let Abe a generalized G7-embedded abelian surface in PV0, then according to the type ofζA=A∩P+2 the surfaceAis
∗The first author was partially supported by the European Commision through HPRN-CT- 2000-00099.
Received May 4, 2004; in revised form December 3, 2004.
type ofζA description (1,1,1,1,1,1) smooth and abelian
(2,1,1,1,1) translation scroll(E,±σ )with2·σ =0 (3,1,1,1) tangent scroll(E,0)
(2,2,2) double translation scroll(E,±σ )with2·σ =0andσ =0 (2,2,1,1) union of seven quadrics
(4,2) union of seven double projective planes (2,2,2)c union of fourteen projective planes
For the translation and tangent scrolls,Eis a septimic(this is the term used by Sylvester), i.e. of degree seven, elliptic curve with an origin and the translation defined by the specified pointσ.
The two distinct(2,2,2)cases (abusively denoted by(2,2,2)and(2,2,2)c) are described in Figure 1 of the appendix.
In the first section we recall some basic facts on(1,7)-polarized abelian surfaces with level structure, and construct a compactification of their moduli space. In fact, we consider a rational map κ : PV0 P6 defined byG7- invariant hypersurfaces of degree 7, which maps any general surface A ∈ A(1,7)to a six-secant plane to a certain Veronese surfaceS⊂P6of degree 9.
The surfaceSis the image byκofP+2, so the six points of intersectionS∩κ(A) is the image ofA∩P+2. It turns out that distinct surfaces Aare mapped to distinct planesκ(A), so the variety of six-secant planes toS form a natural compactification ofA(1,7). The variety of planes inP6that intersectsSin a subscheme of length six is the Fano 3-foldV (K4). Its name originates from the fact that the finite subschemesA∩P+2 form polar hexagons to a certain Klein quartic curve K4 ⊂ P+∗2 , while V (K4) form the compact variety ofapolar subschemes of length six toK4, cf. [17]. It is in this interpretation that Gross and Popescu identifiesV (K4)as a compactification ofA(1,7), cf. [10]. The varietyV (K4)may also be identified with the variety of twisted cubic curves apolar to a certain “Kleinian” net of quadric surfaces. This interpretation is the key to the original approach of Manolache and Schreyer. Although our approach is slightly different from these approaches in the interpretation of V (K4), the main technical argument appears in their papers.
The varietyV (K4)is a prime Fano threefold of genus 12. Mukai discovered different interpretations of these threefolds that are carefully explained in [17].
In the second section we present some useful aspects in our situation of these interpretations. In particular, we describe carefully the subvarietyK4
of V (K4) parameterizing apolar subschemes of length six toK4 which are singular, i.e. do not consist of six distinct points, or equivalently planes inP6 that intersectSin a singular subscheme of length six.
In the last section we prove Theorem 1.1 by considering the inverse images by κ of planes that belong to K4 ⊂ V (K4) as surfaces inPV0. The final argument consists in verifying that only the planes that belong toK4 pull back to singular surfaces.
Note that A. Marini also investigates such degenerations in [14]. His ap- proach uses the interpretation ofV (K4)as the set of twisted cubic curves apolar to the “Kleinian” net of quadrics.
Notations
The base field is the one of complex numbersC. IfR is a vector space, the Veronese map fromRtoSnR(as well as its projectivisation) will be denoted byνn:
R−−−→νn SnR.
Ifs ∈Hilb(n,PR)the type ofs( i.e. the associated length partition ofn) will be labeledλs:
s−−−→λ λs.
IfH is a hypersurface ofPRtheneH =0 is an equation ofH.
The irreducible representations of SL(2,F7)will be denoted byC,W3,W3∨, U4,U4∨, W6,U6,U6∨,W7,W8andU8. The algebra of representations of the group SL(2,F7)is a quotient of
Z[C, W3, W3∨, U4, U4∨, W6, U6, U6∨, W7, W8, U8]
where C denotes the trivial representation, Wn denotes an irreducible PSL(2,F7)-module of dimensionnandUndenotes an irreducible SL(2,F7)- module of dimensionnon which SL(2,F7)actsfaithfully.
The corresponding table of multiplication can be found in [13] and [5] with the following possible identifications
[5] V1 V3=V− V3∗ V4=V+ V4∗ V6 V6 V6∗ V7 V8 V8
[13] I W W U U T T1 T2 L M1 M2
× C W3 W3∨ U4 U4∨ W6 U6 U6∨ W7 U8 W8
· P+2 Pˇ+2 P−3 Pˇ−3 P+5 P−5 Pˇ−5 P+6 P−7 P+7 Note that what are denoted byP+2 andP−3 are respectively denoted byP2−and P3+in [10].
2. Moduli space: a compactification
In this section we describe our main object, the abelian surfaces with a(1,7)- polarization and a level structure, and their moduli spaceA(1,7). The general member ofA(1,7)is embedded inP6 invariant under a groupG7. The hy- persurfaces of degree 7 invariant under this group define a rational map onP6 which is the key to our approach to a compactification of A(1,7). The first analysis of this map is the main aim of this section.
LetAbe an abelian surface, i.e. a projective complex torusC2/where is a (maximal) lattice ofC2 R4. Then the variety Pic0(A)is an abelian surface as well (isomorphic to(C2)∨/∨); this latter one is called the dual abelian surface ofAand will be denoted byA∨. As additive group, the surface Aacts on itself by translation, ifx∈Awe will denote byτxthe corresponding translation.
A line bundle of type(1,7)onAis the data of an ample line bundleL such that the kernel of the isogeny
ϕL :A−−−→A∨, x−−−→τx∗L ⊗L−1 is isomorphic toZ7×Z7.
A(1,7)-polarization onAis an element of
{(A, ϕL)|L is of type(1,7)}.
Thanks to Mumford, a coarse moduli space of(1,7)-polarized abelian sur- faces exists, we will denote it byM(1,7).
Now choose ageneric(1,7)-polarized abelian surface, sayA, then V0 = H0(A,L)is of dimension 7. The group ker(ϕL)Z7×Z7becomes a sub- group ofPSL(V0). It is certainly safer to work with linear representations rather than projective ones so we need to lift the action ofZ7×Z7onPV0to an action of one of its central extensions onV0. The Schur multiplier ofZ7×Z7
is known to beµ7so any projective representation ofZ7×Z7is induced by a linear representation of what is called the “Heisenberg group of level 7” and denoted byH7: that is to say for alln∈N∗and all projective representations ρwe get a Cartesian diagram:
1−−−→µn−−−→SL(n,C)−−−→PSL(n,C)−−−→1
↑ ↑ ↑ρ
1−−−→µ7−−−→ H7 −−−→ Z7×Z7 −−−→1
In this way V0 becomes aH7-module (of rank 7), this representation is called the “Schrödinger” representation ofH7. We now have a way to identify
allthe vector spaces H0(A,L)for any abelian surfaceA∈M(1,7)as they are all isomorphic toV0asH7-modules. This looks too good to be true. So what is wrong? We implicitly made an identification between ker(ϕL)andZ7×Z7 and this is certainly defined up to SL(2,F7)only! So the construction is only invariant underN7=H7SL(2,F7)which turns out to be the normalizer of H7in SL(7,C)SL(V0).
So to any basissof ker(ϕL)corresponds an embedding
#s :A−→PV0.
The group SL(2,F7)acts on the set of bases of ker(ϕL)and we immediately get another complication (which will turn out to be quite nice after all):
#s(A)=#−s(A).
Let us denote byG7 = H7{−1,1} ⊂N7. This group (after killingµ7) is in general the full group of automorphisms of the surface#s(A): ifbis any element ofZ7×Z7andτb : PV0 −→ PV0 is the involution induced by the corresponding “−1” ofG7, thenτb leaves#s(A)(globally) invariant and is induced by the “opposite” mapx → −xonAfor a good choice of the image of the origin on#s(A). In other words,τb·#s =#−s.
As the cardinality of SL(2,F7)/{−1,1} =PSL(2,F7)is 168, each element ofM(1,7)will be mapped intoPV0in 168 ways (distinct in general). We get a brand new moduli space by considering a(1,7)-polarized abelian surface together with one of its embeddings, this moduli space will be denoted by A(1,7):
A(1,7)=
((A, ϕL), s)|(A, ϕL)∈M(1,7), sis a basis of ker(ϕL) / in which the equivalence relationis the expected one,(X1, s1)(X2, s2)if
#s1(X1)= #s2(X2)(fortunately, this impliesX1 =X2). The choice of basis (or embedding) is the level structure referred to in the introduction.
Here are some useful remarks:
(1) The surface#s(A)is of degree 14;
(2) by construction ifx ∈ A, then the set of 49 points#s(ϕL−1(ϕL(x)))is an orbit under the action ofH7(orH7/µ7if we want to be precise);
(3) the above construction works as well for elliptic curves, so in particular PV0contains naturallyG7-invariant embedded elliptic curves (of degree 7);
(4) ifb ∈ Z7×Z7the involution τb induces a SL(2,F7)-module structure onV0, as such a moduleV0splits inV0=W3⊕U4where bothW3and
U4are irreducible SL(2,F7)-modules of dimension 3 and 4 respectively (such thatS3W3∨S2U4). The projective planePW3and the projective spacePU4 in PV0 are point wise invariant by the involution τb. For a givenb∈Z7×Z7these two spaces will often be denoted byP+2 andP3− (the signs come from the following:W3 is also aPSL(2,F7)-module, i.e.−1∈ SL(2,F7)acts trivially on it, but SL(2,F7)acts faithfully on U4);
(5) ifEis aG7-invariant elliptic curve inPV0then the curveEintersects anyP+2 in one point (corresponding to the image of 0) and anyP−3 in three points (corresponding to its non trivial 2-torsion points);
(6) the latter holds also for abelian surfaces, with decomposition 6+10 corresponding to the odd and even 2-torsion points ([12]);
(7) by adding a finite set ofG7-invariant heptagons to the union of theG7- invariant embedded elliptic curves of degree 7, one gets a birational model of the Shioda modular surface of level 7. It intersects eachP+2 in a plane quartic curveK4, the so called Klein quartic curve ([15] or [8]
which contains original references to Klein).
Following what happens in the(1,5)case we consider the rational map κ :PV0P(H0(OPV0(7))G7)∨
i.e. the rational transformation of PV0 by the linear system of G7-invariant septimics. In what follows, by a ‘G7-invariant septimic’ we always mean a septimic in this linear system. Obviously the vector space H0(OPV0(7))G7 is aPSL(2,F7)-module. On the other hand h0(OPV0(7))G7 = 8 (cf. [13]), soκ takes,a priori, its values in aP7. There is a uniqueN7-invariant septimic hyper- surface [15], so the decomposition of thePSL(2,F7)-module H0(OPV0(7))G7 must have a 7-dimensional summand. But the only dimensions of non-trivial irreduciblePSL(2,F7)-modules are 3, 6, 7 and 8, andW7is the only one of dimension 7, so this must be the other summand. Therefore H0(OPV0(7))G7 W7⊕Cas aPSL(2,F7)-module.
We will show show that the image ofκis in fact contained inPW7. First we analyze the base locus of these septimic hypersurfaces.
Lemma2.1.AG7-invariant septimic hypersurface of PV0contains any of the fortynine projective spacesP−3.
Proof. Consider the restriction to any projective spaceP−3 = PU4ofG7- invariant septimic hypersurfaces. Then we get a map
H0(OPV0(7))G7 −→H0(OPU4(7))=S7U4∨
which needs to be SL(2,F7)-equivariant (the entire collection ofP−3’s being invariant under the action ofG7). ButU4 is a faithful module for SL(2,F7) and 7 is odd, so the map is the zero map.
Using Bezout’s theorem we get
Corollary2.2.AG7-invariant septimic hypersurface of PV0contains any G7-invariant elliptic curve of PV0 as well as its translation scroll by a non trivial2-torsion point.
Notice that our forty nineP+2 constitute an orbit underG7, so it makes sense to considerthesurfaceκ(P+2).
Corollary2.3.‘The’ planeP+2 is mapped byκto a Veronese surfaceSof degree nine inPW7.
Proof. The restriction of nG7-invariant septimic hypersurface toP+2 con- tains, by the previous corollary and the last item (7) above, the Klein quartic curveK4. The residual factor is a cubic, so the image of the restriction map H0(OPV0(7))G7 −→ H0(OPW3∨(7)) =S7W3∨factors throughW7 ⊂ S3W3∨ = W7⊕W3. Therefore the restriction ofκtoP+2 is defined byW7⊂S3W3∨which forms a basepoint free linear system of cubics, and the corollary follows.
Remark 2.4. This phenomenon holds also in the (1,5) case where P2+ is mapped by the linear system of G5-invariant quintic hypersurfaces to a (projected) Veronese surface of degree 25 in a Grassmannian Gr(1,P3)⊂ P5 known as the bisecants variety of a certain rational sextic curve in P3. The image of the blow-up of ‘the’ line P−1 is the sextic complex in Gr(1,P3) of lines contained in a dual sextic of planes inP3. In this case, anyG5-embedded (1,5)-polarized abelian surface is mapped to a ten-secant plane to the image ofP+2 (which intersects the sextic complex along six lines).
Although the same kind of results are expected in our situation, here is a difference between the two cases. In the(1,5)case the vector space ofG5- invariant quintics is spanned by determinants of socalled Moore matrices.
Remark2.5. The vector space H0(OPV0(7))G7 is not spanned by determ- inants of (symmetric) Moore matrices ([10]). For this, let us recall first what a Moore matrix is; there is a nice isomorphism of irreducible N7-modules (defined up to homothety)S2V4 = U4⊗V0 which induces a map U4 −→
S2V4⊗V0∨. For a good choice of basis inV0∨we get a 7×7 matrix with coeffi- cients inV0∨which is called a (symmetric) “Moore matrix”. Now considering determinants, i.e. the mapS2V4−−−→“S27” C) we get a first map
S7U4−→S7V0∨=H0(OPV0(7)),
which composed with the projection to the invariant part yields a map S7U4−→H0(OPV0(7))G7.
This latter one is certainly zero: the action of−1∈SL(2,F7)cannot be trivial on any SL(2,F7)-invariant subspace of the vector spaceS7U4.
Neverthelessanti-symmetricMoore matrices play a fundamental role in the (1,7)case. They are defined by the isomorphism of irreducibleN7-modules 2V4=W3∨⊗V0. The locus (inPV0) where such a matrix drops its rank is a Calabi Yau threefold (see [10]) and will appear in subsection 4.3.
Proposition 2.6. The image by the map κ of a G7-embedded (1,7)- polarized abelian surface is (generically) a projective plane and we have a factorization
A&−−−−−→49:1 A∨&−−−−→2:1 KA&∨ −−−−→2:1 κ(A)
where the surfaceA&is the blowup of the surfaceAalong its intersection with the base locus of theG7-invariant septimics, andKA&∨is the quotient ofA∨&
by the involution, i.e. in general its Kummer surface.
Proof. Assume the(1,7)-polarization ofA∈ M(1,7)is given by a very ample line bundle, then from theG7-equivariant resolution of the surfaceAin PV0 which can be found in [13, appendix], one can check that dim(H0(OA(7))G7)=3.
If the mapκ|Ais finite, then we have a factorization A&−−−−−→49:1 A∨&−−−−→2:1 KA&∨ −−−−→2:1 κ(A).
The first two maps (as well as their degree) come from the construction itself, the degree of the last one follows by Bezout’s theorem.
If κ|A is not finite, then it is composed with a pencil. We may assume that Pic(A)has rank 1, i.e. all curves are hypersurface sections or translates thereof. But no such curve isG7-invariant unless the curve is a possible translate of a septimic hypersurface section, soκ|A has at most isolated base points.
Therefore the linear system definingκ|Ais a subsystem of|7·h|. The linear system is a net, so if it is composed with a pencil each member is reducible.
In fact the general member must be the reducible union of seven hyperplane sections through the base locus. The intersection of one of these hyperplane sections with the base locus is a finite set whose stabilizer inG7has order at least 14. Therefore the hyperplane itself must have stabilizer of order at least 14. But there are only finitely many such hyperplanes, so this is impossible.
Thus the mapκis finite onAand the proposition follows.
Let us denote by A(1,7)v the (open) subset of A(1,7) corresponding to (1,7)-polarized abelian surfaces for which the polarization is given by a very ample line bundle andκ(A)is a plane. The association
A→κ(A)
mapsA(1,7)v into the variety of six-secant planes toκ(P+2). Notice that the six pointsκ(A)∩κ(P+2)=κ(A∩P+2). Gross and Popescu in [10] prove that A∩P+2 is a polar hexagon to the Klein quartic curve. On the other hand six points inP+2 form a polar hexagon to the Klein curve precisely if all four cubics in their ideal is contained inW7⊂S3W3∨i.e. when their span on the Veronese surfaceκ(P+2)is a plane. The variety of planes that intersectκ(P+2)in a finite scheme of length six therefore define a natural compactificationA(1,7)v. In this compactification, an abelian surfaceA∈A(1,7)v is the proper transform of a six-secant plane of the Veronese surface byκ−1.
Moreover, any(1,7)-polarized abelian surface is mapped into the hyper- planePW7ofPH0(OPV0(7))G7so their union is contained in a septimic hyper- surface ofPV0. Therefore we have
Corollary2.7. The compactificationA(1,7)vis isomorphic to the unique prime Fano threefold of genus12which admitsPSL(2,F7)as its automorph- isms group. The universal(1,7)-polarized abelian surface with level7struc- ture is birational to the uniqueN7-invariant septimic hypersurface of PV0.
Proof. Let us denote byX7the uniqueN7-invariant septimic hypersurface ofPV0 and byBκ the base locus of theG7-invariant septimic hypersurfaces.
PutY4=κ(X7\Bκ)⊂PW7and consider the diagram I
p1
❅
p2X7\Bκ−−−→κ Y4 G(3, W7)
whereI ⊂PW7×G(3, W7)denotes the graph of the incidence correspondence betweenPW7and the (projective) fibers of the tautological sheaf overG(3, W7) and wherep1andp2are the natural projections. In order to prove birationality we just need to prove that a general point ofX7is contained in one (and only one) abelian surface. One first needs to remark, using representation theory for instance, that both the hypersurfacesX7andY4are irreducible.
LetA∈A(1,7)v aG7-embedded abelian surface. We have:
• the septimic hypersurfaceX7contains the surfaceA;
• the surfaceAintersectsP+2 along a reduced scheme;
• the surfaceAis not contained in the base locusBκ.
The only non obvious fact is the third item. ButBκ intersects P+2 along a Klein quartic curve K4 so if we hadA ⊂ Bκ this would imply the non emptiness ofA∩K4and in such casesA∩P+2 admits a double point (see e.g.
section 3 below) contradicting the second item. Next the mapA−→A∩P2+ is injective (see [10]) so the planeκ(A\Bκ)entirely characterizes the surface A. Summing up we get that two distinct surfacesAandAintersect each other either on
• the threefoldBκ (which is of codimension 2 inX7),
• or on the preimage byκof the points inY4⊂PW7which are contained in more than one six-secant plane to the Veronese surfaceκ(P+2\K4). SinceAis not contained inBκ, it remains to show thatAis not contained in the second locus. But one proves easily that the second locus is 2-dimensional, being the preimage of the union of the Veronese surfaceκ(P+2\K4)itself and its ruled surface of trisecant lines (for which the base is isomorphic to the Klein quartic curveK4of the dual planePˇ+2).
Remark2.8. Notice that one can also prove (using Schubert calculus) that the hypersurfaceY4has degree four inPW7(this is true for any collection of six-secant planes to such a projected Veronese surface).
With this compactification ofA(1,7)v, we define
Definition2.9. AgeneralizedG7-embedded abelian surfaceis the proper transform byκ−1of a plane that intersects the Veronese surfaceκ(P+2)in a finite scheme of length six.
Notice that to each generalizedG7-embedded abelian surfaceAone may associate a subschemeζA=κ(A)∩κ(P+2)of length six.
3. Fano threefoldsV22
The natural boundary of the compactificationA(1,7)vconstructed above con- sists of planes that intersectκ(P+2)in nonreduced subschemes of length six.
The aim of this section is to describe this boundary in terms of the degrees of the components of these subschemes, but first we need some general facts on this compactification as a prime Fano threefold of genus 12 in its anticanonical embedding.
Recall Mukai’s characterization of prime Fano threefolds of genus 12 (cf.
[16]).
Definition-Proposition3.1. Any Fano threefold of index 1 and genus12 is isomorphic to the variety of sums of powers
VSP(F,6)=
(/1, . . . , /6)∈Hilb6PW∨|eF ≡6 i=1e/4i
of a plane quartic curveF. Conversely, ifF is not a Clebsch quartic (i.e. its catalecticant invariant vanishes), then VSP(F,6)is a Fano threefold of index 1 and genus 12. Its anti-canonical model is denoted byV22.
3.1. Construction
LetW be an irreducible SL(3,C)-module of dimension 3, we have a decom- position of SL(3,C)-modules ([9])
S2(S2W)∨=S4W∨⊕S2W generating an exact sequence
0−→S4W∨−→Hom(S2W, S2W∨).
So a plane quarticF in PW whose equation is given by ‘an’ elementeF of S4W∨gives rise to ‘a’ morphismαF :S2W −→ S2W∨and a quadricQF in P(S2W)of equation αF(x)·x = 0 or evenx ·αF(x) = 0 by the canonical identificationS2W =(S2W∨)∨. From the equality h0(Iν2(F )(2))=7, we get acharacterization of this quadricby the two properties:
(i) the two forms onWdefined byαF(ν2(−))·ν2(−)andeFare proportional i.e. the quadricQF and the Veronese surfaceν2(PW)intersect along the image of the plane quarticF underν2;
(ii) the quadricQF is apolar to the Veronese surfaceν2(PW∨)ofPS2W∨i.e.
apolar to each element of the vector space H0(Iν2(PW∨)(2))S2W∨ ⊂ S2(S2W).
Lemma3.2 (Sylvester). The minimal integernfor which VSP(F, n)is non empty is the rank ofαF(called the catalecticant invariant of the quartic curve).
Proof. This well known result of Sylvester (see e.g. Dolgachev and Kanev [6], Elliot [7, page 294]) can be deduced from the following observation: let n∈N∗, then
ν2(VSP(F, n))= {(p1, . . . , pn)∈VSP(QF, n)|p×∈ν2(PW∨)}.
Indeed if s = (/1, . . . , /n) ∈ VSP(F, n) then eF ≡ n
i=1e4/i for a good normalization ofel× and the quadricQ⊂PS2W of equation
eQ≡n
i=1
eν22(/i)
is endowed with the two properties which characterize the quadric QF: the second one is a direct consequence of H0(Iν2(PW∨)(2)) ⊂ H0(Iν2(s)(2))and the first one arises by construction. Applyingν2−1we get the required equality.
Define the vector spaceY/ ⊂ S2W such that the line /of the planePW induces the exact sequence
0−−−→C·e2/ −−−→S2W∨−−−→Y/∨−−−→0, that is to sayY/is the orthogonal space (inS2W) ofe/2.
Definition 3.3. The subscheme C/ of the plane PW∨ defined byC/ = {x∈PW∨|e2x∈αF(Y/)} =ν2−1(αF(Y/))is called theanti-polar conicof the line/(with respect to the quarticF).
Alternatively, ifαF has maximal rank we have obviously C/ = {x∈PW∨|e2x·αF−1(e2/)=0}.
Setn=rank(αF); the construction of a point ofVSP(F, n)is now very easy by the following corollary, which is a consequence of the classical construction of a point ofVSP(Q, n)whenQis a quadric of rankn.
Corollary3.4. A point(/1, . . . , /n)lies in VSP(F, n)if and only if/i ∈ C/j wheni =j,
We turn to the anti-canonical embedding ofV22, in particular to 3.2. Conics on the anti-canonical model
LetV be the seven dimensional vector space defined by the exact sequence 0−−−→W −−−→S3W∨−−−−→pF V −−−→0
where the second map is induced byF ∈S4W∨ ⊂Hom(W, S3W∨)and de- note byV2,9the image ofPW∨inPV by the Veronese embeddingν3composed with the third mappF.
By definitions =(/1, . . . , /6)∈VSP(F,6)if and only if the image bypF
of the 6-dimensional vector space (inS3W∨) spanned bye3/i is of rank 3. Thus we get a map ofVSP(F,6)into the GrassmannianG(3, V ), by(/1, . . . , /6)→ pF(/1, . . . , /6).
Remark 3.5. The image ofVSP(F,6)in the Plücker embedding of the Grassmannian is the anti-canonical model V22 of this Fano threefold, it is isomorphic to the variety of six-secant planes to the projected Veronese surface V2,9.
Now it is reasonable to talk about conics onV22. Denote byF& the dual quartic ofPW of equationα−F1(e2/)·e2/ =0, in other words we have
F&= {/∈PW∨|/∈C/}
(the quarticF&reduces to a double conic whenn=5), and denote byHF the sextic ofPW∨given by
HF = {/∈PW∨|rank(αF−1(e2/))1}.
Let/ ∈ PW∨\HF, then the anti-polar conicC/ is smooth and we can write the abstract rational curve C/ as P1 = PS1 with dimS1 = 2, i.e. we put S1 =H0(OC/(1)). PutSn := SnS1, thenPSnis identified with the divisors of degreenonPS1and we have
Lemma3.6. The set of divisors of degree5on the anti-polar conicC/given by{/+C/∩C/, / ∈ C/}is a projective line inPS5. Thisg51admits a base point if/∈F&.
Proof. LetD be such a divisor. By Corollary 3.4 the divisorDis com- pletely determined by any one of its (sub)-divisor of degree 1, so the variety of such divisors is a curve of first degree inPS5.
Corollary 3.7. The points of VSP(F,6) which contain a given line / describe a conicC/ on the anti-canonical modelV22. The two conicsC/ and C/have the same rank.
Proof. Let / be a point outside HF. Then the image of C/ by ν3 is a rational normal sextic projected by the mappFto a smooth sextic insideP4/ := P(pF(H0(OC/(6))))– we have an injection H0(OC/(6)) ⊂ S3W∨ and it is a simple matter to check H0(OC/(6))∩ker(pF)is of dimension 2, moreover identifying ker(pF)andW we haveP(H0(OC/(6))∩W)=/⊂PW. NowP4/ also contains the image ofν3(/)and projecting from this latter point the sextic becomes:
(i) a rational sextic on a quadric of aP3generically;
(ii) a rational quintic on a quadric of aP3if/∈F&.
These curves are obviously on a quadric, since a six-secant plane toV2,9passing through the pointpF(ν3(/))will be mapped to a five-secant (resp. four-secant) line to this rational curve.
Now if/ ∈ HF,C/ breaks in two lines, say/1 and /2. We get two sys- tems of six-secant planes toV2,9containingpF(ν3(/)), one of these intersects pF(ν3(/1)) in two fixed points and intersects the twisted cubic pF(ν3(/2))
along a pencil of divisors of degree 3. In particular, such collection is mapped to a line byκ.
Corollary3.8. Ifp is not onHF the threefoldp1p−21(Cp)is a quadric cone6pof rank4inPp4. Ifp∈HF the cone6psplits in twoP3’s.
Remark3.9. We already get a first interpretation in terms of abelian sur- faces. PuttingW =W3∨and choosing the uniquePSL(2,F7)-invariant quartic K4 of PW = ˇP+2 for F we get V = W7, V2,9 = κ(P+2), F& = K4. Now ifp ∈ P+2, the proper transform of the quadric cone6p byκ−1 is by [10]
birational to a Calabi Yau threefold, and by the preceding corollary contains – whenCpis smooth –twodistinct pencils of special surfaces: the one induced by the six-secant planes, parameterized byCpand corresponding generically to abelian surfaces, and another one induced by the second ruling (parameterized byCp) of planes of the cone. We think that these last ones are the same as the ones evoked in [10, remark 5.7].
3.3. Boundary for the Klein quartic
The boundary ofV22=VSP(F,6), as the set of nonreduced length six schemes apolar toF, is easily deducible from what follows. But for the general plane quarticF one needs to introduce a covariant ofF, and this would be beyond the subject of this paper. So in this section, we focus on the surfaceF = {s ∈ VSP(F,6)|λs =(16)}when the quarticF admitsPSL(2,F7)as its group of automorphisms.
We start by choosing a faithful embedding 1−→SL(2,F7)−→SL(3,C), so the vector spaceW of our preceding section becomes a SL(2,F7)-module (necessarily irreducible), sayW W3∨and the decompositionS4W3 = C⊕ W6⊕W8allows us to consider theuniquePSL(2,F7)-invariant quarticK4of Pˇ+2 =PW3. Such a quartic is called a Klein quartic and becomes the quarticF of our preceding section. All the quartic covariants ofF are equal toF (when non zero) and the Klein quarticK4⊂P+2 is (by unicity) the quarticF&of the last section.
We will need the classical
Lemma3.10.There is a uniqueSL(2,F7)-invariant even theta characteristic ϑ on the genus3curveK4(resp.K4).
Proof. The existence follows directly by the existence of a SL(2,F7)- invariant injectionW3−→S2U4so that one can illustrateK4as the Jacobian of a net of quadrics (inPU4∨). It is well known that such Jacobian is endowed with an even theta characteristic (cf. [1]). Reciprocally, such a theta characteristic on a curve of genus 3 comes with a net of quadrics and HomSL(2,F7)(W3, S2U) =0
if and only if the four dimensional vector spaceU equalsU4∨ as SL(2,F7)- module.
We have
Proposition3.11. Letp∈K4and(x1, x2, x3)∈K4×K4×K4such that h0(ϑ +p−xi) = 1, then the anti-polar conicCxi ofxi with respect toK4
containsxj.
Proof. Let us leave the planeP+2 and take a look at the configuration in P+5 =PS2W3=PW6= ˇP+5. The image ofxiby the Veronese embeddingν2lies on the quadricQK4. On the other hand, noticing that HomSL2F7(C, S2S2W3)= Cthis quadric can be interpreted
• as the inverse of the quadricQK4;
• as the Plücker embedding of the Grassmannian of lines ofP−3 using the SL(2,F7)-invariant identificationS2W32U4.
Let us denote byK6the Jacobian of the net of quadrics given byW3−→S2U4 and remember that this curve is (by unicity) canonically isomorphic toK4it- self. So ν2(xi) is a line in P−3 (still denoted by ν2(xi)) and this one turns out to be a trisecant line to the sexticK6 containing the image ofp by the identificationK4 = K6. This interpretation of the(3,3)correspondence on K4 = K6 induced by the even theta characteristic as the incidence corres- pondence betweenK6and its trisecant lines is due to Clebsch. Now the three linesν2(xi)are concurrent inp and then the three pointsν2(xi) ofP+5 span a projective plane contained in the inverse of the quadricQK4. In particular, αK−14(ν2(xi))·ν2(xj)=0 which is precisely what we need to claim thatxj ∈Cxi.
Notice that using the same geometric interpretation we get immediately Corollary 3.12.If p ∈ K4 then the anti-polar conic Ca intersects the hessian triangleTp(i.e. the hessian of the polar cubic ofpwith respect toK4) in points of the quarticK4(andCp∩K4−2p=Tp∩K4−2x1−2x2−2x3)).
Proposition3.13. Lets ∈:=K4, then there exists at least one pointp in the support ofζssuch thatp∈K4and the type ofζsis one of the following
p /∈H6 p∈H6
2,1,1,1,1 2,2,1,1 type ofζs 3,1,1,1 4,2
2,2,2 (2,2,2)c
whereH6is the Hessian ofK4.
The types ofζs and the corresponding stratification of is illustrated in Figure 1 and Figure 2 in the appendix.
Proof. From the preceding section, a pointsofVSP(K4,6)is inif and only if the support ofζs intersects the quartic curve K4. So let p ∈ K4, by Corollary 3.4 the only thing to understand is the type ofζs when the pointp moves along the conicCp. We have the alternative: the conic Cp is smooth (case i) orp∈H6:=HK4(case ii).
(i) Denote (once again) by Sn the (n + 1)-dimensional vector space H0(OCa(n)), we haveSn = SnS1. As p ∈ K4, the(1,5) correspond- ence between the two (isomorphic) rational curvesCpandCphas a base point, namely the pointpitself onCpand then reduces to a(1,4)corres- pondence. The induced pencil of divisors of degree 4 inPS4intersects the variety of non reduced divisors in six points (as any generic pencil inPS4) and the expected types ofζs are hence(2,1,1,1,1)generically, (3,1,1,1)once and(2,2,1,1)six times (each corresponding to a point ofCp∩K4− {p}). But by the preceding proposition, ifp ∈Cp∩K4
andp = p, then the two conicsCpandCp intersect inp+p+2p withp∈K4hence the six expected subschemesζs of type(2,2,1,1) onCp become threeζs of type(2,2,2)for the particular Klein quartic.
Notice that in such a case, the schemeζs has a length decomposition 2·(p+p+p)and there exists a pointq ∈K4so that h0(ϑ+q−x)=1 wheneverx ∈ {p, p, p}. Let us denote byqxthe intersection ofCxwith the linexq, then
p3·qp+p3·qp +p3·qp =0 is an equation ofK4.
(ii) Suppose now the pointpis one of 24 points of intersection of the quartic K4and its HessianH6. Such points come 3 by 3 and the groupµ3acts on each triplet (so there is an orderp1, p2, p3on such triplet). Putp=p1. The conicCp1 is no longer smooth and decomposes in two lines, say / = p1p2 and / = p2p3. Each generic point q of the line / gives us a point 2p1 +q +q+ 2p3 = Cp1 ∩Cq + p1 +q of (hence of type(2,2,1,1)) withq ∈ / defined such that the degree 4 divisor p2+p1+q+q on the line/is harmonic. One can even provide the corresponding equation of the quarticK4:
;(βx+αz)4−;(βx−αz)4−2αβ{(x+;(β2z−α2y))4−x4} +2α3β((y+;z)4−y4)=0
