arXiv subjects

Massimiliano Alessandro

Publications and source records attributed to Massimiliano Alessandro.

4 recordsLinked to original sources

arXiv2026

Finite quotients of full surface braid groups and complex surfaces of general type: cyclic, dihedral, and extra-special quotients

Let $\mathsf{B}_2(Σ_g)$ be the full braid group on two strings on a compact Riemann surface of genus $g$. We compute the number of finite cyclic, dihedral and extra-special quotients $φ\colon \mathsf{B}_2(Σ_g) \to G$, under the assumption that the quotient map $φ$ does not factor through $π_1(\operatorname{Sym^2}Σ_g)$. We then apply our algebraic results to the geometric problem of constructing smooth surfaces of general type as Galois covers of $\operatorname{Sym^2}(Σ_g)$ branched on the diagonal. In particular, we construct two $3$-dimensional families of minimal surfaces of general type with $p_g=7$, $q=4$ and $K^2=32$ such that members of different families have the same biregular invariants and the same Betti numbers, but different torsion part for the first homology group.

math.GR↗

arXiv2026

Pluricanonical Geometry of Varieties Isogenous to a Product and Abelian Covers

We study canonical and pluricanonical maps of varieties isogenous to a product of curves, i.e., quotients of the form $X = (C_1 \times \dots \times C_n)/G$ with $g(C_i)\ge 2$ and $G$ acting freely. For this purpose, we provide a technical result which is of general interest: a decomposition theorem for pluricanonical systems of abelian covers. This theorem provides an effective tool for the explicit study of geometric properties, such as base loci and the birationality of pluricanonical maps. For threefolds isogenous to a product, we prove that the 4-canonical map is birational for $p_g \ge 5$ and construct an example attaining the maximal canonical degree for this class of threefolds. In this example, the canonical map is the normalization of its image, which admits isolated non-normal singularities. Computational classifications also reveal threefolds where the bicanonical map fails to be birational, even in the absence of genus-2 fibrations. This illustrates an interesting phenomenon similar to the non-standard case for surfaces.

math.AG↗

arXiv2023

On the components of the Main Stream of the moduli space of surfaces of general type with $p_g=q=2$

We give first an easy construction of surfaces with $p_g=q=2, K^2=5$ and Albanese map of degree $3$, describing an irreducible connected component of the moduli space of surfaces of general type, which we show to be the only one of the Main Stream with these invariants and satisfying a mild condition. We call it the family of CHPP surfaces, since it contains the family constructed by Chen and Hacon, and coincides with the one considered by Penegini and Polizzi. We also give an easy construction of an irreducible connected component of the moduli space of surfaces of general type with $p_g=q=2, K^2=6$ and Albanese map of degree $4$, which we call the family of PP4 surfaces since it contains the family constructed by Penegini and Polizzi. Finally, we answer a question posed by Chen and Hacon, via three families of surfaces with $p_g=q$ whose Tschirnhaus module has a kernel realization with quotient a nontrivial homogeneous bundle. Two families have $p_g=q=3$, the third is a new family of surfaces with $p_g=q=2, K^2=6$ and Albanese map of degree $3$ (the existence of this family is based on arXiv:2212.14877, joint work of the second author with Edoardo Sernesi) .

math.AG↗

arXiv2023

Semi-Projective Representations and Twisted Representation Groups

A semi-projective representation is a homomorphism of a finite group into the group of semi-projective transformations of a finite dimensional vector space over a field. Schur's concept of a representation group for projective representations is extended to semi-projective representations under the assumption that the field is algebraically closed. A computer algorithm is given that produces, for a given finite group, all such twisted representation groups under trivial or conjugation actions on the field of complex numbers.

math.GR↗