arXiv subjects

Benedetta Piroddi

Publications and source records attributed to Benedetta Piroddi.

6 recordsLinked to original sources

arXiv2026

Coble type hypersurfaces and hyperk\"ahler fourfolds

A classical result, already observed by Coble, asserts that a genus 2 Jacobian can be embedded in $\mathbf P^8$ as the singular locus of a unique cubic hypersurface; similarly, the Kummer of a genus 3 Jacobian is embedded in $\mathbf P^7$ as the singular locus of a unique quartic hypersurface. We present a precise analogue of these results in the context of hyperk\"ahler fourfolds: for the general member in the 20-dimensional locally complete families of polarized hyperk\"ahler fourfolds of $\mathrm{K3}^{[2]}$-type with squares 4 or 6 and divisibility 1, we establish the existence of a unique Coble type hypersurface. As a consequence, we describe several geometric aspects of the corresponding moduli spaces.

math.AG↗

arXiv2026

A symplectic fourfold

We present a method to construct irreducible symplectic varieties by studying terminalisations of quotient of hyper-K\"ahler manifolds by non-natural group actions. In particular, we construct irreducible symplectic varieties of dimension $4$ with $b_2 = 4$ and non-quotient singularities: this provides explicit examples of ISVs for which a global Torelli theorem is not known to hold.

math.AG↗

arXiv2024

Symplectic actions of groups of order 4 on K3^[2]-type manifolds, and standard involutions on Nikulin-type orbifolds

Given a K3^[2]-type manifold X with a symplectic involution i, the quotient X/i admits a Nikulin orbifold Y as terminalization. We study the symplectic action of a group G of order 4 on X, such that i belongs to G, and the natural involution induced on Y (the two groups give two different results). We give a lattice-theoretic classification of X and Y in the projective case, and give some explicit examples of models of X. We also give lattice-theoretic criteria that a Nikulin-type orbifold N has to satisfy to admit a symplectic involution that deforms to an induced one.

math.AG↗

arXiv2024

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

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.

math.AG↗

arXiv2023

On the transcendental lattices of Hyperkähler manifolds

We introduce the notion of a Hyper-Kähler manifold $X$ induced by a Hodge structure of K3-type. We explore this notion for the known deformation types of hyper-Kähler manifolds studying those that are induced by a K3 or abelian surface, giving lattice-theoretic criteria to decide whether or not they are birational to a moduli space of sheaves over said surface. We highlight the different behaviors we find for the particular class of hyper-Kähler manifolds of O'Grady type.

math.AG↗

arXiv2022

K3 surfaces with a symplectic automorphism of order 4

Given $X$ a K3 surface admitting a symplectic automorphism $τ$ of order 4, we describe the isometry $τ^*$ on $H^2(X,\mathbb Z)$. Having called $\tilde Z$ and $\tilde Y$ respectively the minimal resolutions of the quotient surfaces $Z=X/τ^2$ and $Y=X/τ$, we also describe the maps induced in cohomology by the rational quotient maps $X\rightarrow\tilde Z,\ X\rightarrow\tilde Y$ and $\tilde Y\rightarrow\tilde Z$: with this knowledge, we are able to give a lattice-theoretic characterization of $\tilde Z$, and find the relation between the Néron-Severi lattices of $X,\tilde Z$ and $\tilde Y$ in the projective case. We also produce three different projective models for $X,\tilde Z$ and $\tilde Y$, each associated to a different polarization of degree 4 on $X$.

math.AG↗