Special subvarieties of EPW sextics
Special subvarieties of EPW sextics
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EPW sextics 的特殊亚种
DOI：
10.1007/s00209-012-0980-5
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发表时间：
2012
影响因子：
0.8
通讯作者：
A. Ferretti
中科院分区：
文献类型：
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作者：
A. Ferretti
我们研究 EPW 六角星的几何形状，以产生特殊的亚品种。特别是，我们展示了一个（奇异）Enriques 曲面，并计算了它在六次幂 Chow 环中的类。为了做到这一点，我们将双 EPW 六次方程显式退化为四次曲面上两点的希尔伯特方案，无论是在光滑情况还是在奇异情况下（保持奇异性）。双 EPW 六次幂上覆盖对合的固定轨迹退化到四次曲线的双切线曲面，只要四次曲线获得足够的节点，就可以证明其对于 Enriques 曲面是双有理的。 Ferretti（代数数论，2009a）中使用此构造作为起点，以证明 Beauville 和 Voisin 在不可约辛簇 Chow 环上的猜想，特别是在非常一般的双 EPW 六次幂的情况下。
We study the geometry of EPW sextics in order to produce special subvarieties. In particular we exhibit a (singular) Enriques surface and we compute its class in the Chow ring of the sextic. In order to do this, we produce an explicit degeneration of double EPW sextics to a Hilbert scheme of two points on a quartic surface, both in the smooth and in the singular case (keeping the singularities). The fixed locus of the covering involution on the double EPW sextic degenerates to the surface of bitangents to the quartic, which can be shown to be birational to an Enriques surface, provided the quartic acquires enough nodes. This construction is used in Ferretti (Algebra Number Theory, 2009a) as a starting point to prove a conjecture of Beauville and Voisin on the Chow ring of irreducible symplectic varieties, in the particular case of a very general double EPW sextic.