Dual Double EPW-sextics and Their Periods
Dual Double EPW-sextics and Their Periods
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双双 EPW-sextics 及其周期
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发表时间：
2006
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通讯作者：
K. O’Grady
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作者：
K. O’Grady
Eisenbud，Popescu和Walter在P中构造了一些具有自然二重覆盖的奇异六次超曲面(EPW-Sextics)([4]的例子(9.3))：我们已经证明了[13]一般的这种二重覆盖是K3的Hilbert平方的变形，并且双EPW-Sextics族是(K3)的局部逆射影变形族。因此，双EPW-性线的族类似于四次三次线上的Fano变种线族(见[2])，但有以下区别：Fano变种上的Plucker充分因子是Beauville-Bogomolov二次型的平方6(见[1，2])，而双EPW性线的自然极化是平方2(见[13])。设Y⊂P是一般的EPW性：我们在[13]中证明了DUAL Y∨⊂(P5)∨是另一个一般的EPW性。因此，我们可以将Y的自然双覆盖X联系到一个“对偶”变种X∨，即Y∨的自然双覆盖。这种构造定义了双EPW-性别的模空间上的一个(有理)对合。在文[13]中，我们证明了一般EPW-六项式不是自对偶的，因此双EPW-六项式的模空间上的对合不是恒等式；在本文中我们确定了双EPW-六项式的周期与其对偶之间的关系。在陈述结果之前，我们先回顾一下EPW-性征的定义。设V是6维C-向量空间。选择一个同构VOL：∧V∼−→C，设ω是∧V上的辛形式，由楔积和VOL定义。设P(V)是V的一维子向量空间的射影空间，则ω给∧V⊗OP(V)一个秩为20的辛向量丛的结构。设F是∧V⊗OP(V)的子向量丛，其纤维F[v]在[v]∈P(V)上由可被v整除的张量组成：
Eisenbud, Popescu and Walter have constructed certain singular sextic hypersurfaces (EPW-sextics) in P (Example (9.3) of [4]) which come provided with a natural double cover: we have shown [13] that the generic such double cover is a deformation of the Hilbert square of a K3 and that the family of double EPWsextics is a locally versal family of projective deformations of (K3). Thus the family of double EPW-sextics is similar to the family of Fano varieties of lines on a cubic 4-fold (see [2]), with the following difference: the Plucker ample divisor on the Fano variety of lines has square 6 for the Beauville-Bogomolov quadratic form (see [1, 2]) while the natural polarization of a double EPWsextic has square 2 (see [13]). Let Y ⊂ P be a generic EPW-sextic: we proved in [13] that the dual Y ∨ ⊂ (P5)∨ is another generic EPW-sextic. Thus we may associate to the natural double cover X of Y a “dual”variety X∨ namely the natural double cover of Y ∨. This construction defines a (rational) involution on the moduli space of double EPW-sextics. In [13] we showed that a generic EPW-sextic is not self-dual and hence the involution on the moduli space of double EPW-sextics is not the identity; in this paper we determine the relation between the periods of a double EPW-sextic and its dual. Before stating the result we recall the definition of EPW-sextics. Let V be a 6-dimensional C-vector space. Choose an isomorphism vol : ∧ V ∼ −→ C and let ω be the symplectic form on ∧V defined by wedge product followed by vol. Let P(V ) be the projective space of 1-dimensional sub vector spaces of V ; then ω gives ∧V ⊗OP(V ) the structure of a symplectic vector-bundle of rank 20. Let F be the sub-vector-bundle of ∧V ⊗OP(V ) whose fiber F[v] over [v] ∈ P(V ) consists of tensors divisible by v: