The Chow ring of double EPW sextics

Abstract

A conjecture of Beauville and Voisin states that for an irreducible symplectic variety X the subring of CH * (X ) generated by divisors goes injectively into the cohomology of X , via the cycle map. We prove this for a very general double Eisenbud-Popescu-Walter sextic.

Moduli of double EPW-sextics

“…Table ( 6) gives the restriction of q ∨ A to q w1 (α), q w2 (α) . The entry on the second line and second column is non-zero by (3.4.29), the others are zero by (3.4.23), thus the second condition of Item (2) is satisfied.…”

“…It follows that the ordering rays are generated by vectors (m, r 1 , r 2 , r 3 ) such that m ∈ {0, ±1} and (r1,r2,r3)∈{(0,0,0), (0,1,0), (6,0,0), (0,0,6), (6,6,0), (0,6,6), (3,0,3), (3,3,3), (3,6,3), (12,6,6), (6,6,12), (4,2,2), (2,2,4)}.…”

“…Now we come to the relation with one of Looijenga's compactifications of complements of hyperplane arrangements. Suppose that A approaches A + generically: then X A will approach the Hilbert square of a quartic K3 surface, see [6]. Similarly if A approaches A k or A h generically then X A will approach the Hilbert square of a K3 of genus 2 or a moduli space of pure sheaves on such a K3.…”

Chow Rings, Decomposition of the Diagonal, and the Topology of Families

“…(i) We compute P 1 • P 1 . From (5.39), (2.5), and the projection formula (2.4), we get that P 1 • P 1 is the morphism Rπ * → Rπ * induced by the cycle class 40) where the p ij are the various projections from X × B X × B X to X × B X. We now use the facts that p *…”

“…We will not comment on the proof of (2). Let us just say that the result in (2) was partially extended by Ferretti in [40] to the case of O'Grady fourfolds (see [77]). …”