Zero-cycles on double EPW sextics

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 Hilbert square of a K3 surface, we show that the action of the anti-symplectic involution on the Chow group of zero-cycles commutes with the Fourier decomposition of Shen–Vial.

Chow rings of Mp̃0,2(N,d) and M¯0,2(ℙN−1,d) and Gromov–Witten invariants of projective hypersurfaces of degree 1 and 2

In this paper, we prove formulas that represent two-pointed Gromov–Witten invariant [Formula: see text] of projective hypersurfaces with [Formula: see text] in terms of Chow ring of [Formula: see text], the moduli spaces of stable maps from genus [Formula: see text] stable curves to projective space [Formula: see text]. Our formulas are based on representation of the intersection number [Formula: see text], which was introduced by Jinzenji, in terms of Chow ring of [Formula: see text], the moduli space of quasi maps from [Formula: see text] to [Formula: see text] with two marked points. In order to prove our formulas, we use the results on Chow ring of [Formula: see text], that were derived by Mustaţǎ and Mustaţǎ. We also present explicit toric data of [Formula: see text] and prove relations of Chow ring of [Formula: see text].