arXiv subjects

Gian Paolo Grosselli

Publications and source records attributed to Gian Paolo Grosselli.

4 recordsLinked to original sources

arXiv2021

Explicit analysis of positive dimensional fibres of $ \mathcal{P}_{g,r} $ and Xiao conjecture

We focus on the positive dimensional fibres of the Prym map $\mathcal{P}_{g,r}$. We present a direct procedure to investigate infinitely many examples of positive dimensional fibres. Such procedure uses families of Galois coverings of the line admitting a 2-sheeted Galois intermediate quotient. Then we generalize to families of Galois coverings of the line admitting a Galois intermediate quotient of higher degree and we show that the higher degree analogue of the aforementioned procedure gives all the known counterexamples to a conjecture by Xiao on the relative irregularity of a fibration.

math.AG↗

arXiv2021

Shimura subvarieties in the Prym locus of ramified Galois coverings

We study Shimura (special) subvarieties in the moduli space $A_{p,D}$ of complex abelian varieties of dimension $p$ and polarization type $D$. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to $\mathbb{P}^1$. We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.

math.AG↗

arXiv2021

Higher dimensional Shimura varieties in the Prym loci of ramified double covers

In this paper we construct Shimura subvarieties of dimension bigger than one of the moduli space of polarised abelian varieties of a given dimension, which are generically contained in the Pym loci of (ramified) double covers. The idea is to adapt the techniques already used to construct Shimura curves in the Prym loci to the higher dimensional case, namely to use families of Galois covers of the projective line. The case of abelian covers is treated in detail, since in this case it is possible to make explicit computations that allow to verify a sufficient condition for such a family to yield a Shimura subvariety of the space polarised abelian varieties of a given dimension.

math.AG↗

arXiv2020

Shimura curves in the Prym loci of ramified double covers

We study Shimura curves of PEL type in the space of polarised abelian varieties $A^δ_p$ generically contained in the ramified Prym locus. We generalise to ramified double covers, the construction done in [10] in the unramified case and in the case of two ramification points. Namely, we construct families of double covers which are compatible with a fixed group action on the base curve. We only consider the case of one-dimensional families and where the quotient of the base curve by the group is ${\mathbb P}^1$. Using computer algebra we obtain 184 Shimura curves contained in the (ramified) Prym loci.

math.AG↗