arXiv subjects

G. P. Pirola

Publications and source records attributed to G. P. Pirola.

9 recordsLinked to original sources

arXiv2024

Simplicity of some Jacobians with many automorphisms

We study an explicit $(2g-1)$-dimensional family of Jacobian varieties of dimension $\frac{d-1}2(g-1)$, arising from quotient curves of unramified cyclic coverings of prime degree $d$ of hyperelliptic curves of genus $g\ge 2$. By using a deformation argument, we prove that the generic element of the family is simple. Furthermore, we completely describe their endomorphism algebra, and we show that they admit a rank $\frac{d-1}2-1$ group of non-polarized automorphisms. As an application of these results, we prove the generic injectivity of the Prym map for étale cyclic coverings of hyperelliptic curves of odd prime degree under some slight numerical restrictions. This result generalizes in several directions previous results on genus 2.

math.AG↗

arXiv2023

Holomorphic 1-forms on the moduli space of curves

Since the sixties it is well known that there are no non-trivial closed holomorphic $1$-forms on the moduli space $\mathcal{M}_g$ of smooth projective curves of genus $g>2$. In this paper, we strengthen such result proving that for $g\geq 5$ there are no non-trivial holomorphic $1$-forms. With this aim, we prove an extension result for sections of locally free sheaves $\mathcal{F}$ on a projective variety $X$. More precisely, we give a characterization for the surjectivity of the restriction map $ρ_D:H^0(\mathcal{F})\to H^0(\mathcal{F}|_{D})$ for divisors $D$ in the linear system of a sufficiently large multiple of a big and semiample line bundle $\mathcal{L}$. Then, we apply this to the line bundle $\mathcal{L}$ given by the Hodge class on the Deligne Mumford compactification of $\mathcal{M}_g$.

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arXiv2023

On the Hessian of cubic hypersurfaces

In this paper, we analyze the Hessian locus associated to a general cubic hypersurface, by describing for every $n$ its singular locus and its desingularization. The strategy is based on strong connections between the Hessian and the quadrics defined as partial derivatives of the cubic polynomial. In particular, we focus our attention on the singularities of the Hessian hypersurface associated to the general cubic fourfold. It turns out to be a minimal surface of general type: its analysis is developed by exploiting the nature of this surface as a degeneracy locus of a symmetric vector bundle map and by describing an unramified double cover, which is constructed in a more general setting.

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arXiv2021

On the dimension of Voisin sets in the moduli space of abelian varieties

We study the subsets $V_k(A)$ of a complex abelian variety $A$ consisting in the collection of points $x\in A$ such that the zero-cycle $\{x\}-\{0_A\}$ is $k$-nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed that $\dim V_k(A) \leq k-1$ and $V_k(A)$ is countable for a very general abelian variety of dimension at least $2k-1$. We study in particular the locus $\mathcal V_{g,2}$ in the moduli space of abelian varieties of dimension $g$ with a fixed polarization, where $V_2(A)$ is positive dimensional. We prove that an irreducible subvariety $\mathcal Y \subset \mathcal V_{g,2}$, $g\ge 3$, such that for a very general $y \in \mathcal Y $ there is a curve in $V_2(A_y)$ generating $A$ satisfies $\dim \mathcal Y \le 2g - 1.$ The hyperelliptic locus shows that this bound is sharp.

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arXiv2016

Isogenies of Jacobians

We prove by means of the study of the infinitesimal variation of Hodge structure and a generalization of the classical Babbage-Enriques-Petri theorem that the Jacobian variety of a generic element of a $k$ codimensional subvariety of $\mathcal M_g$ is not isogenous to a distinct Jacobian if $g>3k+4$. We extend this result to $k=1, g\ge 5$ by using degeneration methods.

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arXiv2012

The Fano normal function

The Fano surface $F$ of lines in the cubic threefold $V$ is naturally embedded in the intermediate Jacobian $J(V)$, we call "Fano cycle" the difference $F-F^-$, this is homologous to 0 in $J(V)$. We study the normal function on the moduli space which computes the Abel-Jacobi image of the Fano cycle. By means of the related infinitesimal invariant we can prove that the primitive part of the normal function is not of torsion. As a consequence we get that, for a general $V$, $F-F^-$ in not algebraically equivalent to zero in $J(V)$ (already proved by van der Geer-Kouvidakis) and, moreover, there is no a divisor in $JV$ containing both $F$ and $F^-$ and such that these surfaces are homologically equivalent in the divisor. Our study of the infinitesimal variation of Hodge structure for $V$ produces intrinsically a threefold $Ξ(V)$ in $\mathbb G$ the Grasmannian of lines in $\mathbb P^4.$ We show that the infinitesimal invariant at $V$ attached to the normal function gives a section for a natural bundle on $Ξ(V)$ and more specifically that this section vanishes exactly on $Ξ\cap F,$ which turns out to be the curve in $F$ parameterizing the "double lines" in the threefold. We prove that this curve reconstructs $V$ and hence we get a Torelli-like result: the infinitesimal invariant for the Fano cycle determines $V$.

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arXiv2006

On the topological index of irregular surfaces

We study the topological index of some irregular surfaces that we call generalized Lagrangian. We show that under certain hypotheses on the base locus of the Lagrangian system the topological index is non-negative. For the minimal surfaces of general type with q=4 and p_g=5 we prove the same statement without any hypotheses. Some similar results for higher dimensional varieties are given.

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arXiv2005

Symmetries, Quotients and Kaehler-Einstein metrics

We consider Fano manifolds M that admit a collection of finite automorphism groups G_1, ..., G_k, such that the quotients M/G_i are smooth Fano manifolds possessing a Kaehler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that M admits a Kaehler-Einstein metric too.

math.DG↗

arXiv2001

The Weierstrass representation for pluriminimal submanifolds

In this note we prove a Weierstrass representation formula for pluriminimal submanifolds of euclidean spaces. We use this formula to produce new families of examples of pluriminimal submanifolds. We also prove that any affine algebraic manifold can be pluriminimally embedded into some euclidean space in a non holomorphic manner.

math.DG↗