EPW-sextics are special 4-dimensional sextic hypersurfaces (with 20 moduli) which come equipped with a double cover. We analyze the double cover of EPW-sextics parametrized by a certain prime divisor in the moduli space. We associate to the generic sextic parametrized by that divisor a K3 surface of genus 6 and we show that the double epw sextic is a contraction of the Hilbert square of the K3. This result has two consequences. First it gives a new proof of the following result of ours: smooth double EPW-sextics form a locally complete family of hyperkaehler projective deformations of the Hilbert square of a K3. Secondly it shows that away from another prime divisor in the moduli space the period map for double EPW-sextics is as well-behaved as it possibly could be.
A locally complete family of projective symplectic 4-folds is provided by natural double covers of Eisenbud-Popescu-Walter (EPW) sextic hypersurfaces in projective 5-space. The dual of an EPW sextic is another EPW sextic, thus to a double EPW sextic X we may associate a "dual" double EPW sextic X^{\vee}. We show how to obtain the Hodge structure on H^2(X^{\vee}) from that on H^2(X); it turns out that they are isomorphic over the rationals but not over the integers (in general).
An EPW-sextic is a special 4-dimensional hypersurfaces of degree 6 which comes equipped with a double cover which generically is a Hyperkaehler 4-fold deformation equivalent to the Hilbert square of a K3 surface. The family of EPW-sextics is analogous to the family of cubic 4-fold hypersurfaces, more precisely double EPW-sextics are analogous to varieties of lines on cubic 4-folds. This first paper in a series on moduli and periods of double EPW-sextics is mainly concerned with the classification of EPW-sextics which are analogous to cubic 4-folds whose singular locus has strictly positive dim…
Motivated by the Beauville-Voisin conjecture about Chow rings of powers of $K3$ surfaces, we consider a similar conjecture for Chow rings of powers of EPW sextics. We prove part of this conjecture for the very special EPW sextic studied by Donten-Bury et alii. We also prove some other results concerning the Chow groups of this very special EPW sextic, and of certain related hyperkähler fourfolds.
A conjecture of Beauville and Voisin states that for an irreducible symplectic variety X, any polynomial relation between classes of divisors and the Chern classes of X which holds in cohomology already holds in the Chow groups. We verify the conjecture for a very general double EPW sextic.
The Chow rings of hyperKähler varieties are conjectured to have a particularly rich structure. In this paper, we focus on the locally complete family of double EPW sextics and establish some properties of their Chow rings. First we prove a Beauville-Voisin type theorem for zero-cycles on double EPW sextics; precisely, we show that the codimension-4 part of the subring of the Chow ring of a double EPW sextic generated by divisors, the Chern classes and codimension-2 cycles invariant under the anti-symplectic covering involution has rank one. Second, for double EPW sextics birational to the Hilb…
O'Grady showed that certain special sextics in $\mathbb{P}^5$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.
This note is about the Hilbert square $X=S^{[2]}$, where $S$ is a general $K3$ surface of degree $10$, and the anti-symplectic birational involution $ι$ of $X$ constructed by O'Grady. The main result is that the action of $ι$ on certain pieces of the Chow groups of $X$ is as expected by Bloch's conjecture. Since $X$ is birational to a double EPW sextic $X^\prime$, this has consequences for the Chow ring of the EPW sextic $Y\subset{\mathbb{P}}^5$ associated to $X^\prime$.
We study the indeterminacy locus of the period map for double EPW-sextics. We recall that double EPW-sextics are parametrized by lagrangian subspaces of the third wedge-product of a 6-dimensional complex vector-space. The indeterminacy locus is contained in the set of lagrangians containing a decomposable vector. The projectivization of the 3-dimensional support of such a decomposable vector contains a degeneracy subscheme which is either all of the plane or a sextic curve. We show that the period map is regular on any lagrangian A such that for all decomposables in A the corresponding degener…
We prove that the Hilbert square $S^{[2]}$ of a very general primitively polarized K3 surface S of degree $d(n) = 2(4n^2 + 8n + 5)$, $n \geq 1$ is birational to a double Eisenbud-Popescu-Walter sextic. Our result implies a positive answers, in the case when $r$ is even, to a conjecture of O'Grady: On the Hilbert square of a very general K3 surface of genus $r^2 + 2$, $r \geq 1$ there is an antisymplectic involution. We explicitly give this involution on $S^{[2]}$ in term of the corresponding EPW polarization on it.
In analogy to the case of cubic fourfolds, we discuss the conditions under which the double cover $\tilde{Y}_A$ of the EPW sextic hypersurface associated to a Gushel-Mukai fourfold is birationally equivalent to a moduli space of (twisted) stable sheaves on a K3 surface. In particular, we prove that $\tilde{Y}_A$ is birational to the Hilbert scheme of two points on a K3 surface if and only if the Gushel-Mukai fourfold is Hodge-special with discriminant $d$ such that the negative Pell equation $\mathcal{P}_{d/2}(-1)$ is solvable in $\mathbb{Z}$.
We show that the Hilbert scheme of two points on the Vinberg $K3$ surface has a 2:1 map onto a very symmetric EPW sextic $Y$ in $\mathbb{P}^5$. The fourfold $Y$ is singular along $60$ planes, $20$ of which form a complete family of incident planes. This solves a problem of Morin and O'Grady and establishes that $20$ is the maximal cardinality of such a family of planes. Next, we show that this Hilbert scheme is birationally isomorphic to the Kummer type IHS fourfold $X_0$ constructed in [DW]. We find that $X_0$ is also related to the Debarre-Varley abelian fourfold.
We study the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of \bigwedge^3{\mathbb C}^6 by the natural action of SL_6, call it M. This is a compactification of the moduli space of smooth double EPW-sextics. We determine the stable points, the irreducible components of the GIT boundary and their dimensions. There are strong analogies with the moduli space of cubic 4-folds: we prove a result which is analogous to a theorem of Laza stating that cubic 4-folds with simple singularities are stable. Our final goal (not achieved in the present paper) is to understand th…
We explain a general construction of double covers of quadratic degeneracy loci and Lagrangian intersection loci based on reflexive sheaves. We relate the double covers of quadratic degeneracy loci to the Stein factorizations of the relative Hilbert schemes of linear spaces of the corresponding quadric fibrations. We give a criterion for these double covers to be nonsingular. As applications of these results, we show that the double covers of the EPW sextics obtained by our construction give O'Grady's double EPW sextics and that an analogous construction gives Iliev-Kapustka-Kapustka-Ranestad'…
Eisenbud Popescu and Walter have constructed certain special 4-dimensional sextic hypersurfaces as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation of the Hilbert square of a K3-surface and that the family of such varieties is locally complete for deformations that keep the hyperplane class of type (1,1) - thus we get an example similar to that (discovered by Beauville and Donagi) of the Fano variety of lines on a cubic 4-fold. Conversely suppose that X is an irreducible symplectic 4-fold numerically equivalent to the Hilbert square of…
We prove that the invariant locus of the involution associated to a general double EPW sextic is a constant surface and introduce a filtration on $\CH_1$ of a Gushel-Mukai fourfold. We verify the sheaf/cycle correspondence for sheaves supported on low degree rational curves, parallel to the cubic fourfolds case.
A simple sextic is a reduced complex projective plane curve of degree 6 with only simple singularities. We introduce a notion of Z-splitting curves for the double covering of the projective plane branching along a simple sextic, and investigate lattice Zariski k-ples of simple sextics by Z-splitting curves. We define specialization of lattice types, and classify all lattice types of Z-splitting curves of degree less than or equal to 3 up to specializations.
In this paper we study the automorphisms group of some $K3$ surfaces which are double covers of the projective plane ramified over a smooth sextic plane curve. More precisely, we study the case of a $K3$ surface of Picard rank two such that there is a rational curve of degree $d$ which is tangent to the sextic in $d$ points.
Let $X$ be a double cover of $\mathbb P^3$ branched along a sextic surface $Y$. In this paper, we show that, for general $X$, the Abel-Jacobi map associated to the normalization $\tilde F(X)$ of the surface $F(X)$ of curves contained in $X$ which are preimages of lines bitangent to $Y$, gives an isogeny between the Albanese variety of $\tilde F(X)$ and the intermediate Jacobian of $X$.
We prove non-rationality and birational super-rigidity of a Q-factorial double cover X of P^3 ramified along a sextic surface with at most simple double points. We also show that the condition #|Sing(X)| < 15 implies Q-factoriality of X. In particular, every double cover of P^3 with at most 14 simple double points is non-rational and not birationally isomorphic to a conic bundle. All the birational transformations of X into elliptic fibrations and into Fano 3-folds with canonical singularities are classified. We consider some relevant problems over fields of finite characteristic. When X is de…
In previous work, we have introduced a program aimed at studying the birational geometry of locally symmetric varieties of Type IV associated to moduli of certain projective varieties of K3 type. In particular, a concrete goal of our program is to understand the relationship between GIT and Baily-Borel compactifications for quartic K3 surfaces, K3's which are double covers of a smooth quadric surface, and double EPW sextics. In our first paper (arXiv:1607.01324), based on arithmetic considerations, we have given conjectural decompositions into simple birational transformations of the period ma…
Let SU_X(3) be the moduli space of semi-stable vector bundles of rank 3 and trivial determinant on a curve X of genus 2. It maps onto P^8 and the map is a double cover branched over a sextic hypersurface called the Coble sextic. In the dual P^8 there is a unique cubic hypersurface, the Coble cubic, singular exactly along the abelian surface of degree 1 line bundles on X. We give a new proof that these two hypersurfaces are dual. As an immediate corollary, we derive a Torelli-type result.
A $K3$ surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple $(X,H)$ with $X$ a deformation of $(K3)^{[n]}$ and $H$ an ample divisor of square 2 for Beauville's quadratic form. If $n=2$ then according to the conjecture $X$ is a double cover of a (singular) sextic 4-fold in $\PP^5$. It follows from the conjecture that a deformation of $(K3)^{[n]}$ carrying a divisor (not necessarily ample) of degree 2 has an anti-symplectic birational involution. We test the conjecture…
In this paper we study the automorphisms group of some K3 surfaces which are double covers of the projective plane ramified over a smooth sextic plane curve. More precisely, we study some particlar case of a K3 surface of Picard rank two.
Given a degree one del Pezzo surface with canonical singularities, the linear series generated by twice the anti-canonical divisor exhibits the surface as the double cover of the quadric cone branched along a sextic curve. It is natural to ask if this description extends to the boundary of a compactification of the moduli space of degree one del Pezzo surfaces. The goal of this paper is to show that this is indeed the case. In particular, we give an explicit classification of the boundary of the moduli space of anti-canonically polarized broken del Pezzo surfaces of degree one as double covers…
In this paper we consider double covers of the projective space in relation with the problem of extensions of varieties, specifically of extensions of canonical curves to $K3$ surfaces and Fano 3-folds. In particular we consider $K3$ surfaces which are double covers of the plane branched over a general sextic: we prove that the general curve in the linear system pull back of plane curves of degree $k\geq 7$ lies on a unique $K3$ surface. If $k\leq 6$ the general such curve is instead extendable to a higher dimensional variety. In the cases $k=4,5,6$, this gives the existence of singular index …
It is one of the wonderful ``coincidences'' of the theory of finite groups that the simple group G of order 25920 arises as both a symplectic group in characteristic 3 and a unitary group in characteristic 2. These two realizations of G yield two G-covers of the moduli space of configurations of six points on the projective line modulo PGL_2, via the 3- and 2-torsion of the Jacobians of the double and triple cyclic covers of P^1 branched at those six points. Remarkably these two covers are isomorphic. This was proved over C by transcendental methods by Hunt and Weintraub. We give an algebraic …
Let $S_{g}$ be a general prime K3 surface in $P^g$ of genus $g \geq 3$ or a general double cover of $P^2$ ramified along a sextic curve for $g = 2$ and $S = S_{i,g}$ its {\it i}-th Veronese embedding. In this article we compute the corank of the Gaussian map $Φ:\bigwedge^2 H^0(S,O_S(1)) \to H^0(S,Ω_S^1(2))$ for $i \geq 2, g \geq 2$ and $i=1, g \geq 17$. The main idea is to reduce the surjectivity of $Φ$ to an application of the Kawamata-Viehweg vanishing theorem on the blow-up of $S \times S$ along,the diagonal. This is seen to apply once the hyperplane divisor of the K3 surface $S$ can be dec…
