EPW Cubes

EPW cubes

ByAtanas Ilievat Seoul,Grzegorz Kapustkaat Kraków, Michał Kapustkaat Stavanger andKristian Ranestadat Oslo Dedicated to Piotr Pragacz on the occasion of his 60th birthday

Abstract. We construct a new20-dimensional family of projective six-dimensional ir- reducible holomorphic symplectic manifolds. The elements of this family are deformation equivalent with the Hilbert scheme of three points on a K3 surface and are constructed as natural double covers of special codimension-three subvarieties of the GrassmannianG.3; 6/.

These codimension-three subvarieties are defined as Lagrangian degeneracy loci and their con- struction is parallel to that of EPW sextics, we call them the EPW cubes. As a consequence we prove that the moduli space of polarized IHS sixfolds ofK3-type, Beauville–Bogomolov degree4and divisibility2is unirational.

1. Introduction

By an irreducible holomorphic symplectic (IHS)2n-fold we mean a2n-dimensional sim- ply connected compact Kähler manifold with trivial canonical bundle that admits a unique (up to a constant) closed non-degenerate holomorphic two-form and is not a product of two man- ifolds (see [3]). The IHS manifolds are also known as hyperkähler and irreducible symplectic manifolds, in dimension2they are calledK3surface.

Moduli spaces of polarizedK3surfaces are a historically old subject, studied by the clas- sical Italian geometers. Mukai extended the classical constructions and proved unirationality results for the moduli spacesM_2d parametrising polarizedK3surfaces of degree2d for many cases withd 19(see [19, 21, 23]). On the other hand it was proven in [8] thatM_2d is of general type for d > 61and some smaller values. Note that when the Kodaria dimension of such moduli space is positive, the generic element of such moduli space is believed to be non-constructible.

There are only five known descriptions of the moduli space of higher dimensional IHS manifolds (all these examples are deformations equivalent toK3^Œn). In dimension four we have the following unirational moduli spaces:

A. Iliev was supported by SNU grant 0450-20130016, G. Kapustka by NCN grant 2013/08/A/ST1/00312, M. Kapustka by NCN grant 2013/10/E/ST1/00688, and K. Ranestad by RCN grant 239015.

double EPW sextics with Beauville–Bogomolov degreeq D2(see [24]),

Fano scheme of lines on four-dimensional cubic hypersurfaces withq D6(see [4]),

VSP.F; 10/whereF define a cubic hypersurface of dimension4withq D38(see [13]),

zero locus of a section of a vector bundle onG.6; 10/withq D22described in [6].

Moreover, there is only one more known family in dimension8 withq D 2 studied in [17].

Analogously to the case ofK3surfaces there are results in [9] about the Kodaira dimension of the moduli spaces of polarized IHS fourfolds ofK3^Œ2-type: In particular, it is proven that such moduli spaces with split polarization of Beauville–Bogomolov degreeq 24 are of general type (and forq D18or22are of positive Kodaira dimension). We expect that the number of constructible families in higher dimension becomes small.

According to O’Grady [24], the20-dimensional family of natural double covers of spe- cial sextic hypersurfaces inP⁵ (called EPW sextics) gives a maximal dimensional family of polarized IHS fourfold deformation equivalent to the Hilbert scheme of two points on aK3sur- face (this is a maximal dimensional family sinceb₂.S^Œ2/D23forSaK3surface). Our aim is to perform a construction parallel to that of O’Grady to obtain a unirational20-dimensional family (also of maximal dimension) of polarized IHS sixfolds deformation equivalent to the Hilbert scheme of three points on aK3surface (i.e. ofK3^Œ3type). The elements of this fam- ily are natural double covers of special codimension-three subvarieties of the Grassmannian G.3; 6/that we call EPW cubes.

Let us be more precise. LetW be a complex six-dimensional vector space. We fix an isomorphismj W ^⁶W !Cand the skew symmetric form

(1.1) W ^³W ^³W !C; .u; v/7!j.u^v/:

We denote byLG.10;^³W / the variety of ten-dimensional Lagrangian subspaces of^³W with respect to. For any three-dimensional subspaceU W, the ten-dimensional subspace

TU WD ^²U ^W ^³W

belongs toLG.10;^³W /, andP.TU/is the projective tangent space toG.3; W /P.^³W / atŒU .

For anyŒA2LG.10;^³W /andk2N, we consider the following Lagrangian degen- eracy locus, with natural scheme structure (see [28]):

D^A_k D®

ŒU 2G.3; W /jdimA\TU k¯

G.3; W /:

For the fixedŒA 2 LG.10;^³W / we call the scheme D₂^A an EPW cube. We prove that if A is generic then D₂^A is a sixfold singular only along the threefold D₃^A and that D₄^A is empty. Moreover,D₃^A is smooth such that the singularities ofD₂^A are transversal ¹₂.1; 1; 1/

singularities alongD₃^A.

Before we state our main theorem we shall need some more notation. The projectivized representation^³ of PGL.W / on^³W splitsP¹⁹ D P.^³W / into a disjoint union of four orbits:

P¹⁹D.P¹⁹nW /[.F n/[.nG.3; W //[G.3; W /;

where G.3; W /  F P¹⁹, dim./ D 14, Sing./ D G.3; W /, dim.F / D 18, Sing.F / D , see [7]. We call the invariant sets G; ; F andP¹⁹ the (projective) orbits of

^³for PGL.6/. See [16, Appendix] for some results about the geometry ofand its relations with EPW sextics. For any nonzero vectorw 2W, denote by

F_ŒwD hwi ^.^²W / the ten-dimensional subspace of^³W, such that

[

Œw2P.W /

P.F_Œw/DP.^³W /:

We follow the notation of O’Grady [26]:

†D®

ŒA2LG.10;^³W /jP.A/\G.3; W /6D ;¯

;

D®

ŒA2LG.10;^³W /j 9w2WWdimA\FŒw3¯ : We also consider a third subset

€ D®

A2LG.10;^³W /j 9ŒU 2G.3; W /WdimA\TU 4¯ : All three subsets†,,€ are divisors (see [26] and Lemma 3.6). Hence,

LG¹.10;^³W /WDLG.10;^³W /n.†[€/

is a dense open subset ofLG.10;^³W /. Our main result is the following.

Theorem 1.1. If ŒA 2 LG¹.10;^³W /, then there exists a natural double cover YA

of the EPW cubeD₂^A branched along its singular locusD₃^A such that YA is an IHS sixfold ofK3^Œ3-type with polarization of Beauville–Bogomolov degreeq D 4 and divisibility2. In particular, the moduli space of polarized IHS sixfolds of K3^Œ3-type, Beauville–Bogomolov degree4and divisibility2is unirational.

We prove the theorem in Section 5 at the very end of the paper. The plan of the proof is the following: In Proposition 3.1 we prove that forŒA 2 LG¹.10;^³W /, the varietyD₂^Ais singular only along the locusD₃^A, and that it admits a smooth double coverYA!D^A₂ branched alongD₃^Awith a trivial canonical class. The proof of the Proposition is based on a general study of Lagrangian degeneracy loci contained in Section 2. By globalizing the construction of the double cover to the whole affine varietyLG¹.10;^³W /, we obtain a smooth family

Y!LG¹.10;^³W /

with fibersY_ŒA DYA. Note that the familyYis naturally a family of polarized varieties with the polarization given by the divisors defining the double cover.

In Lemma 3.7 we prove that n.€ [†/ is nonempty. Following [26, Section 4.1], we associate to a generalŒA0 2 n.€ [†/a K3 surfaceSA₀. Then, in Proposition 4.1, we prove that there exists a rational two-to-one map from the Hilbert schemeS_A^Œ3

0 of length-3 subschemes onSA0 to the EPW cubeD₂^A0. We infer in Section 5 that in this case the sixfold Y_A0 is birational toS_A^Œ3

0. Together with the fact thatY_A0 is smooth, irreducible and has trivial canonical class, this proves thatYA₀is IHS.

Since flat deformations of IHS manifolds are still IHS, the familyYis a family of smooth IHS sixfolds. The fact that the obtained IHS manifolds are ofK3^Œ3-type is a straightforward consequence of Huybrechts theorem [12, Thm. 4.6].

During the proof of Theorem 1.1 we retrieve also some information on the constructed varieties. We prove in Section 2.3 that the polarization giving the double coverYA ! D₂^A has Beauville–Bogomolov degreeq./D4and is primitive. Moreover, the degree of an EPW cubeD^A₂ G.3; 6/P¹⁹is480.

Let us recall that the coarse moduli spaceMof polarized IHS sixfolds ofK3^Œ3-type and Beauville–Bogomolov degree4has two components distinguished by divisibility. We conclude the paper by proving that the image of the moduli mapLG¹.10;^³W /!Mdefined byYis a20-dimensional open and dense subset of the component ofMcorresponding to divisibility2 (see Proposition 5.3).

Acknowledgement. The authors wish to thank Olivier Debarre, Alexander Kuznetsov and Kieran O’Grady for useful comments, O’Grady in particular for pointing out a proof of Proposition 5.3.

2. Lagrangian degeneracy loci

In this section we study resolutions of Lagrangian degeneracy loci. Let us start with fixing some notation and definitions. We fix a vector spaceW2n of dimension2nand a symplectic form! 2 ^²W_2n. LetX be a smooth manifold and letW DW2nO_X be the trivial bundle with fiberW2nonX equipped with a non-degenerate symplectic form!Q induced on each fiber by!. Consider a Lagrangian vector subbundleJ W, i.e. a subbundle of ranknwhose fibers are isotropic with respect to!. LetQ A W2n be a Lagrangian vector subspace inducing a trivial subbundleAW. For eachk2Nwe define the set

D_k^AD®

x2X jdim.Jx\Ax/k¯ X;

whereJx andAx denote the fibers of the bundlesJ andAas subspaces in the fiberWx. Let us now defineLG!.n; W2n/to be the Lagrangian Grassmannian parameterizing all subspaces ofW2nwhich are Lagrangian with respect to!. ThenJ defines a mapWX !LG!.n; W2n/ in such a way that J D L, where L denotes the tautological bundle on the Lagrangian GrassmannianLG!.n; W2n/. Moreover, similarly as onX, we can define

D^A_k D®

ŒL2LG!.n; W2n/jdim.L\A/k¯

LG!.n; W2n/;

which admits a natural scheme structure as a degeneracy locus. We then have D_k^AD ¹D_k^A;

i.e. the scheme structure onD_k^Ais defined by the inverse image of the ideal sheaf ofD_k^A; see [11, p.163].

2.1. Resolution ofD_k^A. For eachk2N, letG.k; A/be the Grassmannian ofk-dimen- sional subspaces ofAand let

DQ^A_k D®

.ŒL; ŒU /2LG!.n; W2n/G.k; A/jLU¯ :

By [28],DQ_k^Ais a resolution ofD_k^A. We shall describe the above variety more precisely. First of all we have the following incidence described more generally in [28]:

DQ_k^A

~~

##

D_k^A G.k; A/:

The projection is clearly birational, whereas is a fibration with fibers isomorphic to a Lagrangian GrassmannianLG.n k; 2n 2k/. In particular, DQ^A_k is a smooth manifold of Picard number two with Picard group generated by H, the pullback of the hyperplane sec- tion of LG.n; W2n/ in its Plücker embedding, and R, the pullback of the hyperplane sec- tion ofG.k; A/ in its Plücker embedding. Denote byQ the tautological bundle on G.k; A/

seen as a subbundle of the trivial symplectic bundleW2n˝O_G.k;A/. Consider the subbundle Q^? W2n˝O_G.k;A/ perpendicular toQ with respect the symplectic form. The following was observed in [28].

Lemma 2.1. The varietyDQ_k^Ais isomorphic to the Lagrangian bundle F WDLG.n k;Q^?=Q/:

Of course the tautological Lagrangian subbundle onLG.n k;Q^?=Q/can be identified with the bundleL=QDWW. In particular, we have

c1.W/Dc1.L/ c1.Q/DR H:

Lemma 2.2. The relative tangent bundleT ofWF !G.k; A/is the bundleS².W^_/.

Proof. This can be seen by globalizing the construction of the tangent space of the Lagrangian Grassmannian described for example in [22].

Lemma 2.3. The canonical class ofDQ_k^Ais .nC1 k/H .k 1/R.

Proof. We use the exact sequence

0!T !TF !TG.k;A/ !0:

NowW^_has rankn k, so

c1.T/Dc1.S².W^_//D.nC1 k/c1.W^_/D.nC1 k/.H R/

whilec1.T_G.k;A//DnR. Hence

K_F D c1.T_F/D .nC1 k/H .k 1/R:

Lemma 2.4. The varietyD₁^Ais a hyperplane section ofLG!.n; W2n/.

Proof. Indeed D^A₁ is the intersection of the codimension-one Schubert cycle on the GrassmannianG.n; 2n/ with the Lagrangian Grassmannian, hence a hyperplane section of the Lagrangian Grassmannian.

Let us denote byEthe exceptional divisor of. Lemma 2.5. ForkD2we haveŒEDŒH  2ŒR.

Proof. It is clear thatŒE DaŒH CbŒRfor somea; b 2 Z. Let us now consider the restriction ofEto a fiber of, i.e. we fix a vector spaceV2 Aof dimension2and consider LG.n 2; V₂^?=V2/. SinceED ¹D^A₃, we have

E\ ¹ŒV2D®

ŒL2LG.n 2; V₂^?=V2/jdim.L=V2\A=V2/1¯ :

It is hence a divisor of typeD₁^A=V2which is a hyperplane section of the fiber by Lemma 2.4. It follows thataD1.

To compute the coefficient atŒRwe fix a subspaceVn 2of dimensionn 2inAand consider the Schubert cycle

V_{n 2} D®

ŒU 2G.2; A/jdim.U \Vn 2/1¯ :

The classŒ_{Vn 2}in the Chow group ofG.2; A/is then the class of a hyperplane section. We now describe.V_{n 2}/as the class of the Schubert cyclen 2;nonLG.n; 2n/defined by

n 2;n D®

ŒL2LG.n; 2n/jdim.L\Vn 2/1;dim.L\A/2¯ : By [28, Theorem 2.1] we have

Œn 2;nDc1.L^_/c3.L^_/ 2c4.L^_/ and

ŒD₂^ADc1.L^_/c2.L^_/ 2c3.L^_/:

In terms of intersection onDQ₂^Athe two equations give

H^{n.n2C1/ 3}\ŒDQ₂^ADc1.L^_/^{n.n2C1/ 2}c2.L^_/ 2c1.L^_/^{n.n2C1/ 3}c3.L^_/ and

H^{n.n2C1/ 4}R\ŒDQ^A₂Dc1.L^_/^{n.n2C1/ 3}c3.L^_/ 2c1.L^_/^{n.n2C1/ 4}c4.L^_/:

Since we know thatEis contracted by the resolution toD₃^A, we also have EH^{n.n2C1/ 4}D0:

We can now computeb:

0DEH^{n.n2C1/ 4} (2.1)

D.H CbR/H^{n.n2C1/ 4} DH^{n.n2C1/ 3}CbH^{n.n2C1/ 4}R

Dc1.L^_/^{n.n2C1/ 4} c1.L^_/²c2.L^_/C.b 2/c1.L^_/c3.L^_/ 2bc4.L^_/ :

Using the theorem of Hiller and Boe on relations in the Chow ring of the Lagrangian Grass- mannian (see [27, Theorem 6.4]), we get

c1.L^_/² D2c2.L^_/ and c2.L^_/²D2 c3.L^_/c1.L^_/ c4.L^_/ : Substituting in (2.1), we get

0D.bC2/deg c1.L^_/c3.L^_/ 2c4.L^_/

D.bC2/degn 2;n: It follows thatb D 2.

2.2. The embedding ofG.3; W /into LG.10;^³W /. LetW be a six-dimensional vector space. LetG DG.3; W /P.^³W /be the Grassmannian of three-dimensional sub- spaces inW in its Plücker embedding. Now, recall for eachŒU 2G,

TU D ^²U ^W ^³W:

The projective spaceP.TU/is tangent toG.3; W /atŒU . LetT be the corresponding vector subbundle of^³W˝OG. LetAbe a ten-dimensional subspace of^³W isotropic with respect to the symplectic formdefined by (1.1) and such thatP.A/\G.3; W /D ;. Recall that for kD1; 2; 3; 4we defined

D_k^AD®

ŒU 2G jdim.T_U \A/k¯ G:

Observe thatT is a Lagrangian subbundle of^³W ˝OGwith respect to the two-form. It follows that we are in the general situation described at the beginning of Section 2, with nD10,W20D ^³W,X DG,J DT andADA. ThenT defines a map

WG.3; W /!LG.10;^³W /; ŒU 7!ŒT_U:

We denote byCU WDP.TU/\G.3; W /the intersection ofG.3; W /with its projective tangent space ŒU . Then C_U is linearly isomorphic to a cone over P² P² with vertex ŒU . The quadrics containing the coneCU plays in this situation a similar role in the local analysis of the singularities ofD_k^Aas the Plücker quadrics containing the GrassmannianP.F_Œw/\G.3; W / in [26]; this will be made more precise in Lemma 2.7.

We aim at proving the following result.

Proposition 2.6. LetA 2 LG.10;^³W /such thatP.A/\G.3; W / D ;. The map is an embedding and.G.3; W //meets transversely all lociD_k^AnD_k^A_C1 fork D1; 2; 3. In particular, eachD_k^Ais of expected dimension.

For the proof we shall adapt the idea of [26] to our context. Let us first describemore precisely locally around a chosen pointŒU02G.3; W /. For this, we choose a basisv1; : : : ; v6

forW such thatU0 D hv1; v2; v3i and defineU₁ D hv4; v5; v6i. For anyŒU  2 G.3; W / we haveT_U D ^²U ^W, soT_U0, T_U1 are two Lagrangian spaces that intersect only at0:

TU₀\TU₁ D0. By appropriate choice ofv4; v5; v6we can also assume thatTU₁\AD0.

Let

V D®

ŒL2LG.10;^³W /jL\TU₁ D0¯ :

The decomposition ^³W D TU0 ˚TU₁ into Lagrangian subspaces and the isomorphism TU₁ ! T_U^_

0 induced by allow us to view a Lagrangian space Lin V as the graph of a symmetric linear mapQ_LWT_U0 !T_U1 DT_U^_

0. Letq_L2Sym²T_U^_

0 be the quadratic form corresponding toQL. The mapŒL7!qLdefines an isomorphismV !Sym²T_U^_

0. Consider the open neighborhood

UD®

ŒU 2G.3; W /jTU \TU₁ D0¯

ofŒU0inG.3; W /. ForŒU  2 Uwe denote byQU WDQT_U andqU WDqT_U the symmetric linear map and the quadratic form corresponding to the Lagrangian spaceTU.

We shall describeqU in local coordinates. Observe that for anyŒU 2G.3; W /, TU \TU₁ D0 $ U \U₁ D0

and that any such subspaceU is the graph of a linear mapˇU WU0 !U₁. In particular, there is an isomorphism

WU!Hom.U0; U₁/; ŒU 7!ˇU

whose inverse is the map

˛ 7!ŒU_˛WD

.v₁C˛.v₁//^.v₂C˛.v₂//^.v₃C˛.v₃//

:

In the given basis.v1; v2; v3/; .v4; v5; v6/forU0andU₁we letBU D.bi;j/i;j2¹1;2;3ºbe the matrix of the linear mapˇ_U. In the dual basis we let.m0; M /, withM D.mi;j/_{i;j2¹1;2;3º}, be the coordinates in

T_U^_

0 D.^³U0˚ ^²U0˝U₁/^_D.^³U0˚Hom.U0; U₁//^_:

Note, that under our identification the mapWG.3; W /!LG.10;^³W /restricted toUis the mapŒU 7!q_U, which justifies our slight abuse of notation in the following lemma.

Lemma 2.7. In the above coordinates, the map

WU3ŒU 7!qU WDqT_U 2Sym²T_U^_

0

is defined by

qU.m0; M /D X

i;j2¹1;2;3º

bi;jM^i;j Cm0

X

i;j2¹1;2;3º

B_U^i;jmi;j Cm²₀detBU;

whereM^i;j,B_U^i;j are the entries of the matrices adjoint toM andB_U. Proof. We write in coordinates the map

^³U0˚ ^²U0˝U₁! ^³U₁˚ ^²U₁˝U0

whose graph is^³U ˚ ^²U ˝U₁whereU is the graph of the mapU0 !U₁ given by the matrixBU.

Let nowQAbe the symmetric mapTU0 !TU₁ DT_U^_

0 whose graph isAand letqAbe the corresponding quadratic form. In this way

D^A_l \UD®

ŒU 2UjdimT_U \A/l¯ D®

ŒU 2Ujrk.Q_U Q_A/10 l¯

; henceD_l^Ais locally defined by the vanishing of the.11 l/.11 l/minors of the1010 matrix with entries being polynomials inbi;j.

First we show that the space of quadrics that defineCU surjects onto the space of quadrics on linear subspaces inP.T_U/.

Lemma 2.8. IfP P.TU/nG.3; 6/is a linear subspace of dimension at most2, then the restriction maprP WH⁰.P.TU/;I_CU.2// !H⁰.P;OP.2//is surjective.

Proof. We may restrict to the case whenP is a plane. SinceC_U P.T_U/\G.3; 6/is projectively equivalent to the cone overP²P²in its Segre embedding, it suffices to show that ifP P⁸is a plane that does not intersectP²P² P⁸, then the Cremona transformation Cr onP⁸defined by the quadrics containingP²P²mapsP to a linearly normal Veronese surface. Note that the ideal ofP²P² P⁸is defined by22minors of a33matrix with linear forms in P⁸, whereas the secant of P² P² is defined by the determinant of this matrix. Since the first syzygies between the generators of this ideal are generated by linear ones we infer from [1, Proposition 3.1] that they define a birational map. Moreover, this Cremona transformation contracts the secant determinantal cubic hypersurface V3 to a to a variety linearly isomorphic to P² P², so the inverse Cremona is of the same kind.

Furthermore, the fibers of the mapV3!P²P²are three-dimensional linear spaces spanned by quadric surfaces inP² P². Now, by assumption, P does not intersect P² P², so the restriction CrjP is a regular, hence finite, morphism. Since the fibers of the Cremona transformation are linear,P intersects each fiber in at most a single point, so the restriction CrjP is an isomorphism. Thus, if Cr.P / is not linearly normal, the linear span hCr.P /i is aP⁴, being a smooth projected Veronese surface. Assume this is the case. Then Cr.P /is not contained in any quadric. Since the quadrics that define the inverse Cremona map Cr.P /to the planeP, these quadrics form only a net when restricted to the four-dimensional spacehCr.P /i. In fact, the complement ofP² P² \ hCr.P /i inhCr.P /i is mapped to P by the inverse Cremona transformation. ThereforehCr.P /imust be contained in the cubic hypersurface that is contracted by this inverse Cremona. Since this hypersurface is contracted to the original P² P², we infer that P is contained in P² P². This contradicts our assumption and concludes our proof.

Lemma 2.9. LetK D A\TU₀ D kerQA TU₀ and assume thatk D dimK 3.

Then for anyl kthe tangent coneC_A;U^l

0 ofD^A_l \UatU0is linearly isomorphic to a cone over the corank-l locus of quadrics inP.H⁰.P.K/;O_P.K/.2///.

Proof. We follow the idea of [25, Proposition 1.9]. If we choose a basis ƒ of T_U^_

0, the symmetric linear mapQ_U is defined by a symmetric matrixM^ƒ.B_U/with entries being polynomials in.bi;j/_{i;j2¹1;2;3º}.

The linear summands of each entry in M^ƒ.BU/ form a matrix that we denote by N^ƒ.BU/. SinceQ0 D0, the entries ofM^ƒ.BU/have no nonzero constant terms. Moreover, by using Lemma 2.7 andƒ0 D .m0; M /, we see that the mapU 3 U 7! q⁰_U 2 Sym²T_U^_

0,

where q_U⁰ is the quadratic form corresponding to the symmetric map defined by the matrix N^ƒ0.BU/, mapsU linearly onto the linear system of quadrics containing the coneCU0. Of course, this surjection is independent of the choice of the basis.

We now choose a basis ƒ in TU₀ in which QA is represented by a diagonal matrix R_k Ddiag¹0; : : : ; 0; 1; : : : ; 1ºwithkzeros in the diagonal. Then

D_l^A\UD®

ŒU 2Ujdim.TU \A/l¯ D®

ŒU 2Ujdim ker.Q_U Q_A/l¯ D®

ŒU 2Ujrk.M^ƒ.BU/ R_k/10 l¯ :

HenceD^A_l is defined in coordinates.bi;j/i;j2¹1;2;3ºonUby.11 l/.11 l/minors of the ma- trixM^ƒ.B_U/ R_k. Furthermore, sinceŒU0is the point0in our coordinates.bi;j/_{i;j2¹1;2;3º}, the tangent cone toD_l^A\UatŒU0is defined by the initial terms of the.11 l/.11 l/

minors ofM^ƒ.BU/ R_k. Note that we can write

M^ƒ.BU/ R_k D R_kCN^ƒ.BU/CZ.BU/;

where the entries of the matrixZ.B_U/are polynomials with no linear or constant terms. We illustrate this decomposition as follows:

0 B B B B B B B

@

N_k^ƒCZ_k

N_1;k^ƒ

C1CZ_1;k^ƒ

C1 : : : N_1;10^ƒ CZ_1;10^ƒ ::

: : :: ::: N_k;kC1^ƒ CZ_k;kC1^ƒ : : : N_k;10^ƒ CZ_k;10^ƒ N_k^ƒ

C1;1CZ_k^ƒ

C1;1 : : : N_k^ƒ

C1;kCZ_k^ƒ

C1;k

::

: : :: ::: N_10;1^ƒ CZ_10;1^ƒ : : : N_10;k^ƒ CZ_10;k^ƒ

1CN_k^ƒ

C1;kC1CZ_k^ƒ

C1;kC1 : : : N_k^ƒ

C1;10CZ_k^ƒ

C1;10

::

: : :: :::

N_10;k^ƒ

C1CZ^ƒ_10;k

C1 : : : 1CN_10;10^ƒ CZ_10;10^ƒ

1 C C C C C C C A :

Let ˆbe an .11 l/.11 l/ minor ofM^ƒ.BU/ Rk and consider its decomposition ˆ D ˆ0C C ˆr into homogeneous parts ˆ_d of degreed. Observe that ˆ_d D 0 for d k l, moreoverˆ_{k lC1}can be nonzero only if the submatrix associated to the minorˆ contains all nonzero entries ofR_k. In the latter caseˆ_{k lC1} is a.kC1 l/.kC1 l/

minor of thekkupper left corner submatrixN_k^ƒ.BU/of the matrixN^ƒ.BU/. Let us now denote byq⁰_U the quadric corresponding to the matrixN^ƒ.B_U/and by^N the mapU 7!q_U⁰ . Then, by changingˆ, we get that the tangent cone ofD_l^A\Uis contained in

CO_A;U^l ₀ WD®

ŒU 2Ujrk.N_k^ƒ.B_U//k l¯ D®

ŒU 2Ujrk.q_U⁰ jK/k l¯ : The latter is the preimage byrK ı^N of the corank-l locus in the projective space of quadrics P.H⁰.P.K/;O_P.K/.2///.

By Lemma 2.8, we have seen thatrKı^N is a linear surjection. So we conclude thatCO_A;U^l

0

is a cone over the corank-l locus of quadrics inP.H⁰.P.K/;O_P.K/.2///with vertex a linear space of dimension10 k.kC1/=2. It follows thatCO_A;U^l

0 is an irreducible variety of codi- mensionl.lC1/=2equal to the codimension ofD^A_l . Thus we have equalityC_A;U^l

0 D OC_A;U^l

0

which ends the proof.

Corollary 2.10. If A is a Lagrangian space in ^³W such that P.A/ does not meet G.3; W /, then the varietyD_l^Ais smooth of the expected codimensionl.lC1/=2outsideD_l^A_C1. Moreover, ifl D2 anddimA\TU0 D 3, i.e.ŒU0is a point inD₃^AnD₄^A, then the tangent coneC_A;U²

0 is a cone over the Veronese surface inP⁵centered in the tangent space ofD^A₃.

Proof of Proposition2.6. It is clear from Lemma 2.7 thatis a local isomorphism into its image, and by Corollary 2.10, the subscheme D_A^k D ¹..G.3; W //\D_A^k/ is smooth outsideD_A^kC1, so.G.3; W //meets the degeneracy loci transversally.

2.3. Invariants. We shall compute the classes of the Lagrangian degeneracy loci D_k^AG.3; W /in the Chow ring ofG.3; W /. We consider the embedding

WG.3; W /!LG.10;^³W /

defined by the bundle of Lagrangian subspacesT on G.3; W /. According to [28, Theorem 2.1] the fundamental classes of the Lagrangian degeneracy lociD_k^Aare

ŒD^A₁DŒc1.T^_/\G.3; W /; ŒD₂^ADŒ.c2c1 2c3/.T^_/\G.3; W /

and

ŒD^A₃DŒ.c1c2c3 2c₁²c4C2c2c4C2c1c5 2c₃²/.T^_/\G.3; W /:

TheP⁹-bundleP.T/is the projective tangent bundle onG.3; W /. SoT^_ fits into an exact sequence

0!G.3;W /.1/!T^_ !OG.3;W /.1/!0 and we get

degD₁^AD168; degD₂^AD480; degD₃^AD720:

Remark 2.11. This may be compared with the degree of the line bundle2H 3E on S^Œ3, whereS is a K3 surface of degree 10, H is the pullback of the line bundle of degree 10on S, and E is the unique divisor class such that the divisor of non-reduced subschemes inS^Œ3 is equivalent to2E. The degree, i.e. the value of the Beauville–Bogomolov form, is q.2H 3E/D4, and the degree and the Euler–Poincaré characteristic of the line bundle are

.2H 3E/⁶D15q.2H 3E/³D960 and .2H 3E/D10:

So if the map defined byj2H 3Ejis a morphism of degree2, the image would have degree 480, likeD^A₂.

In Section 4, we show thatS^Œ3for a generalK3surfaceSof degree10admits a rational double cover of a degeneracy locusD₂^A. However that double cover is not a morphism.

3. The double cover of an EPW cube

Proposition 3.1. LetŒA2LG.10;^³W /. IfP.A/\G.3; W /D ;andD₄^AD ;, then D₂^A admits a double coverf W YA ! D₂^Abranched overD₃^A withYAa smooth irreducible manifold having trivial canonical class.

Before we pass to the construction of the double cover let us observe the following.

Lemma 3.2. Under the assumptions of Proposition3.1the varietyD₂^Ais integral.

Proof. We know thatD^A₂ is of expected dimension. Observe now that by Corollary 2.10 the varietyD₂^Ais irreducible if and only if it is connected. To prove connectedness we perform a computation in the Chow ring of the Grassmannian G.3; W /showing that the class ŒD^A₂ does not decompose into a sum of nontrivial effective classes in the Chow groupA³.G.3; W //

whose intersection is the zero class inA⁶.G.3; W //. More precisely we compute ŒD^A₂D16h³ 12hs2C12s3;

wherehis the hyperplane class onG.3; W /,s2ands3are the Chern classes of the tautological bundle onG.3; W /. We then solve in integer coordinatesa; b; c2Zthe equation

.ah³ bs2Ccs3/ .16 a/h³ .12 b/s2C.12 c/s3 D0

in the Chow groupA⁶.G.3; W //which is generated bys₂³, h³s1s2,s₃². Multiplying out the equation in the Chow ring and extracting coefficients at the generators, we get a system of three quadratic diophantine equations ina; b; c:

(3.1)

8 ˆ<

ˆ:

5a²C4ab b²C56a 20bD0;

6a²C8ab 2b² 4acC2bcC72a 52bC20cD0;

6a² 6abCb²C2ac c² 72aC36b 4c D0:

The only integer solutions are.0; 0; 0/and.16; 12; 12/. This ends the proof.

The plan of the construction of the double cover in Proposition 3.1 is the following. We consider the resolutionDQ^A₂ !D₂^Awith exceptional divisorE. We prove thatE is a smooth even divisor, and hence that there is a smooth double coverYQ ! QD^A₂ branched overE. Finally, we contract the branch divisor of the double cover using a suitable multiple of the pullback of a hyperplane class onD₂^Aby the resolution and the double cover.

Thus, we start by defining the incidences DQ₂^AD®

.ŒU ; ŒU⁰/2G.3; W /G.2; A/jTU U⁰¯ and

DQ^A₂ D®

.ŒL; ŒU⁰/2LG.10;^³W /G.2; A/jLU⁰¯ : They fit in the following diagram:

G.3; W / ^//LG!.10;^³W /

D₂^A

j_DA

2 //

D^A₂

DQ₂^A

˛

OO

Q

//DQ^A₂:

OO

Lemma 3.3. Under the assumptions of Proposition3.1 the varietyDQ₂^A as well as the exceptional locusE of the map˛ are smooth. In particular,˛ is a resolution of singularities ofD^A₂.

Proof. Since we know thatD^A₄ D ;, the resolution˛ W QD₂^A !D^A₂ is just the blow-up ofD₂^AalongD^A₃. Now,DQ₂^AnEis isomorphic toD₂^AnD^A₃, so, by Corollary 2.10, we deduce thatDQ₂^Ais smooth outside E. Letp 2 E QD^A₂. Then˛.p/ 2 D₃^A. TakeP1;P2;P3 to be three general hyperplanes passing through˛.p/. ConsiderZ_P DD₂^A\P1\P2\P3and its strict transformZQ_P QD₂^A. We have the following diagram:

ZQ_P

˛P

// QD^A₂

˛

Z_P ^//D₂^A:

The map˛_P W QZ_P ! Z_P is the blow-up ofZ_P inD₃^A\P1\P2\P3, which by Corollary 2.10 is a finite set of isolated points. By the assumption onP1;P2;P3 the strict transform ZQP contains the whole fiber˛ ¹.p/and hence also p 2 QZP. LetPQi be the strict transform ofPi for i D 1; 2; 3. ThenPQi is a Cartier divisor on DQ₂^A and ZQ_P D QP1 \ QP2 \ QP3 is a complete intersection of Cartier divisors onDQ₂^A. Now, from Corollary 2.10, the exceptional divisorE_P DE\ QZ_P of˛_P is isomorphic to a finite union of disjoint.P²/, one for each point inD^A₃ \P1\P2\P3. ButEP is itself a Cartier divisor onZQP by general properties of the blow-up. ThereforeZQ_P is smooth. We conclude thatDQ₂^Ais smooth atpand similarly, thatE is smooth atp.

We compute the first Chern class of the normal bundle of the embeddingQW QD₂^A! QD^A₂. Lemma 3.4. One has

c₁.QN_Q.DQA

2/j QD^A₂/Dc₁.˛N_{.G.3;W //jLG.10;^}3W //D38h;

where h is the pullback via the resolution ˛ of the restriction of the hyperplane class on G.3; W /toD₂^A.

Proof. From the transversality (Proposition 2.6) we have Q

N_Q.DQA

2/j QD₂^A D˛N_{.G.3;W //jLG.10;^}³_{W /}; which gives the first equality.

To get the second, consider the exact sequence

0!TG.3;W /!.T_LG.10;^³_{W /}/!.N_{.G.3;W //jLG.10;^}³_{W /}/!0;

and observe that

.T_LG.10;^3W //D.S²L^_/DS².L^_/DS²T^_;

where L denotes, as before, the tautological bundle on the Lagrangian Grassmannian LG.10;^³W /. We obtain

c1.˛N_{.G.3;W //jLG.10;^}3W //D 11˛c1.T/ 6h:

Now, from

0!O_{G.3;W /}. 1/!T !T_{G.3;W /}. 1/!0 we obtain˛c1.T/D 4h, which proves the lemma.

Note that in our notation we have Q

H D Qc1.L^_/D˛c1.L^_/D˛c1.T^_/D4h:

We aim now at constructing a double covering ofDQ₂^Abranched alongE. It is enough to prove thatE is an even divisor. This follows from the exact sequence

0!T_DQA

2 ! QT_DQA

2 ! QN_Q.DQA

2/j QD₂^A !0 and Lemma 2.3. Indeed, from them we infer

c1.T_DQA

2/D Q.9H CR/ 38hD Q.R/ 2h;

which, by Lemma 2.5, means

E DE\ QD^A₂ D Q.H 2R/D2K_DQA 2:

By Lemma 3.3 there hence exists a smooth double coverfQ W QY ! QD₂^A branched along the exceptional locus E of the resolution˛. Moreover, from the adjunction formula for double covers we getK_YQ D Qf ¹.E/DW QE.

We now need to contractEQ D Qf ¹.E/onYQ. For that, with slight abuse of notation, we denote byhthe class of the hyperplane section onD₂^AG.3; W /. Thenj Qf˛hjis a globally generated linear system whose associated morphism defines˛ı Qf and hence contractsE to a threefold and is two-to-one onYQ n Qf ¹.E/. It follows by standard arguments (for example applying Stein factorization and [10, Proposition 4.4]) that there exists a numbernsuch that the systemjnfQ˛hjdefines a morphism˛Q W QY !Y which is a birational morphism contracting exactlyEQ to a threefold Zand such that its imageY is normal. We then have the following diagram:

YQ

Q

˛

fQ //DQ₂^A

˛

Y ^{f //}D₂^A;

in whichY admits a two-to-one mapf WY !D₂^Abranched alongD₃^A.

Proof of Proposition3.1. We have constructedY, a normal variety admitting a two-to- one mapf W Y ! D₂^A branched alongD₃^A. ClearlyK_YQ D QE implies KY D 0. It hence remains to prove thatY is smooth. Since˛Q is a contraction that contracts onlyE, it is clearQ thatY is smooth outside ofZ D Q˛.E/. Let nowQ p 2 Zand letp⁰ Df .p/. We then choose three general hypersurfacesP1;P2;P3of degreeninP.^³W /passing throughp⁰. Consider

Z_P DD₂^A\P1\P2\P3 and Z⁰_P DD₃^A\P1\P2\P3:

ThenZ_P⁰ is a finite set of points that includesp⁰. Consider the following natural restriction of the above diagram:

YQ_P

Q

˛_P

fQP //ZQ_P

˛P

YP

f_P //ZP:

Here˛_P D ˛j˛ ¹.Z_P/ W QZ_P ! Z_P is just the blow-up ofZ_P along Z_P⁰ . The exceptional divisorEP is then, by Corollary 2.10, isomorphic to a finite set of disjoint planes that each have normal bundleO_P2. 2/inZQ_P. Taking the double cover ofZQ_P branched along the exceptional divisorE_P, the preimages of these.P²/are the components of EQ_P QY_P, each component a P²with normal bundleO_P². 1/. The contraction˛Q_P contracts the divisorEQ_P to a finite set of points inYP. It contracts one of its.P²/, denote it byEQ_P^p, to the pointp. Note also that from the construction,Y_P is the intersection of three Cartier divisors onY which is smooth outside the finite set of pointsZ_P⁰ . Thus, since we constructedY to be normal, we deduce that Y_P is also normal. We claim thatp must be a smooth point ofY_P. Indeed, we know that˛Q_P is a birational morphism onto the normal varietyYP. Moreover, all linesl QE_P^p D P² are numerically equivalent onYQ_P and satisfy

lK_YQ

P D 1 < 0:

It follows from [18, Corollary 3.6] that there exists an extremal rayrforYQ_P whose associated contraction contr W QY_P ! OY_P contractsEQ_P^p to a pointpO and that˛Q_P factorizes through contr. By [18, Theorem 3.3] we have that contr is the blow-down ofEQ_P^p andpO is a smooth point of YOP. Let us now denote by W OYP !YP the morphism satisfying˛QP D ıcontr. Consider the restrictionoofto small open neighborhoods ofpO andp. Thenois a birational proper morphism which is bijective to an open subset of the normal varietyY_P. It follows by Zariski’s main theorem thatois an isomorphism and in consequence,pis a smooth point onY_P.

The latter implies thatY must also be smooth atp as it admits a smooth complete inter- section subvariety which is smooth atp.

Corollary 3.5. LetŒA 2 LG.10;^³W / be a general Lagrangian subspace with a three-dimensional intersection with some spaceF_ŒwD ¹w^˛j˛ 2 ^²Wº. Then there exists a double coverfA W YA ! D₂^Abranched overD₃^A, whereYAis a smooth irreducible sixfold with trivial canonical class.

Proof. It is enough to make a dimension count to prove that the general Lagrangian spaceAsatisfying the assumptions of the corollary also satisfies the assumptions of Proposi- tion 3.1. Indeed, as in the introduction, let

D®

ŒA2LG.10;^³W /j 9w2WWdim.A\FŒw/3¯

;

€ D®

ŒA2LG.10;^³W /j 9U 2G.3; W /Wdim.A\TU/4¯ : Lemma 3.6. The set€ LG.10;^³W /is a divisor.

Proof. Let us consider the incidence

„D®

.ŒU ; ŒA/2G.3; W /LG.10;^³W /jdim.TU \A/4¯ :

The dimension of„can be computed by looking at the projection„ ! G.3; 6/. For a fixed tangent plane we choose first aP³ inside: this choice has24parameters. Then for a fixedP³ we have dim.LG.6; 12// D 21 parameters for the choice ofA. Thus the dimension of „is 9C24C21D54. It remains to observe that the projection„!LG.10;^³W /is finite, and that dim.LG.10;^³W //D55.

Note that in [26, Proposition 2.2] it is proven thatis irreducible and not contained in

† D ¹ŒA 2 LG.10; 20/ j P.A/\G.3; W / ¤ ;º. Our corollary is now a consequence of Proposition 3.1 and the following lemma.

Lemma 3.7. The divisors,€ LG.10;^³W /have no common components.

Proof. We need to prove dim.\€/ < 54 which, by the fact that is irreducible and not contained in†, is equivalent to dim..\€/n†/ < 54. For this, observe that if ŒA2 .\€/n†then there existŒU 2 G.3; W /andŒw2 P.W /with dim.A\TU/D4 and dim.A\F_Œw/D3. We can hence consider the incidence

‚D®

.ŒA; ŒW3; ŒW4; Œw; ŒU /jW3DA\FŒw; W4 DA\TU

¯ LG.10;^³W /G.3;^³W /G.4;^³W /P.W /G.3; W /

such that its projection to LG.10;^³W / contains.\€/n†. Note also that if we take .ŒA; ŒW3; ŒW4; Œw; ŒU /2‚thenW4\W3DW4\FŒwDW3\TU.

We shall now compute the dimension of‚by considering fibers under subsequent pro- jections:

LG.10;^³W /G.3;^³W /G.4;^³W /P.W /G.3; W /

1

!G.3;^³W /G.4;^³W /P.W /G.3; W /

2

!G.4;^³W /P.W /G.3; W /

3

!P.W /G.3; W /:

We have two possibilities for pairs.Œw; ŒU /which give us two types of points to consider:

(i) Ifw62U, then dimTU \F_ŒwD3.

(ii) Ifw2U, then dimTU \F_ŒwD7.

We then have different types of elements in the intersection₃¹.Œw; ŒU /\2.1.‚//, de- pending on the number

d1WDdim.W4\F_Œw/Ddim.W4\W3/3:

IfW₄^?denotes the orthogonal toW₄with respect toin^³W, then dimW₄^?\F_ŒwD6Cd₁. Now, in order forŒW3to be an element of₂¹.ŒW4; Œw; ŒU /\1.‚/we must have

W₃W₄^?\F_Œw:

The fiber₁¹.ŒW3; ŒW4; Œw; ŒU /\‚is of dimension.3Cd1/.4Cd1/=2. Hence to compute the dimension of each component of‚it is enough to compute the dimensions of the spaces F_i;d1 of elements.ŒW3; ŒW4; Œw; ŒU /of types.i; d1/, wherei D1ifw 62 U andi D2if w2U.

(i) Fori D1we start with a choice ofŒU 2G.3; W /. ThenŒwbelongs to an open subset ofP⁵. We haved13andŒW4belongs to the Schubert cycle consisting of four-spaces in the ten-dimensional spaceTU that meet the fixed three-spaceTU \F_Œwin dimension d1. AndŒW3belongs to the Schubert cycle of three-spaces in the.6Cd1/-dimensional spaceW₄^?\FŒwthat contains the spaceW4\FŒwof dimensiond1