K3 surfaces with a symplectic action of $(\mathbb Z/2\mathbb Z)^2$

Benedetta Piroddi

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arXiv · 2408.00643

K3 surfaces with a symplectic action of $(\mathbb Z/2\mathbb Z)^2$

Benedetta Piroddi

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Abstract

We study the symplectic action of the group (Z/2Z)^2 on a K3 surface X: we describe its action on H^2(X,Z) and the maps induced in cohomology by the rational quotient maps; we give a lattice-theoretic characterization of the resolution of singularities of the quotient X/i, where i is any of the involutions in (Z/2Z)^2. Assuming X is projective, we describe the correspondence between irreducible components of its moduli space, and those of the resolution of singularities of its quotients: this being the first description of this correspondence for a non-cyclic action, we see new phenomena, of which we provide explicit examples assuming $X$ has a polarization of degree 4.

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Benedetta Piroddi. 2024-08-01. K3 surfaces with a symplectic action of $(\mathbb Z/2\mathbb Z)^2$. https://arxiv.org/abs/2408.00643

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