arXiv subjects

Maximiliano Leyton-Alvarez

Publications and source records attributed to Maximiliano Leyton-Alvarez.

3 recordsLinked to original sources

arXiv2025

${\mathbb Z}_{p}^{m}$-actions of type $(d;p,n)$

A ${\mathbb Z}_{p}^{m}$-action of type $(d;p,n)$, where $2 \leq d \leq m \leq n$ are integers, is a pair $(S,N)$ where $S$ is a $d$-dimensional compact complex manifold, $N \cong {\mathbb Z}_{p}^{m}$ is a group of holomorphic automorphisms of $S$ such that the quotient orbifold $S/N$ is the $d$-dimensional projective space ${\mathbb P}^{d}$ whose branch locus consists of $n+1$ hyperplanes in general position, each one of branch order $p$. If $(d;p,n) \notin \{(2;2,5),(2;4,3)\}$ and $d+1 \leq n$, then we prove that: (i) $N$ is a normal subgroup of ${\rm Aut}(S)$ and (ii) if $(S,M)$ is a ${\mathbb Z}_{\hat{p}}^{\hat{m}}$-action of type $(d;\hat{p},\hat{n})$, then $M=N$. If, moreover, $d+1 \leq n \leq 2d-1$, then we observe that $S$ is not algebraically hyperbolic

math.AG↗

arXiv2024

Automorphisms of Generalized Fermat manifolds

Let $d \geq 1$, $k \geq 2$ and $n\geq d+1$ be integers. A $d$-dimensional smooth complex algebraic variety $M$ is called a generalized Fermat variety of type $(d;k,n)$ if there is a Galois holomorphic branched covering $π:M \to {\mathbb P}^{d}$, with deck group $H\cong {\mathbb Z}_{k}^{n}$, whose branch divisor consists of $n+1$ hyperplanes in general position, each one of branch order $k$. In this case, $H$ is called a generalized Fermat group of type $(d;k,n)$. In previous work, we proved that the generalized Fermat group $H$ is unique in the following cases: (i) $d=1$ and $(k-1)(n-1)>2$, or (ii) $d \geq 2$ and $(d;k,n) \notin \{(2;2,5), (2;4,3)\}$. To obtain this uniqueness fact, we used a differential method due to Kontogeorgis. This paper provides a different and shorter proof of the uniqueness of $H$. We also study the locus of fixed points of subgroups of $H$.

math.AG↗

arXiv2015

Familles d'espaces de m-jets et d'espaces d'arcs

Families of m-jet spaces and arc spaces. Let V be an algebraic variety defined over an algebraically closed field of characteristic zero. The m-jet spaces and the arc space provide the information on the geometry of the variety V, therefore it is natural to ask the following questions: When will a deformation of V induce a deformation of the m-jet spaces and the arc space? If one considers a deformation of V which admits a flat simultaneous resolution, how will the image of the Nash application vary? In section 3 some partial answers to these questions can be found. In section 4 two families of hypersurfaces of the 4-dimensional affine space are shown in which the Nash application is bijective. What's more, in each case a desingularization in which all the irreducible components of the exceptional fiber are essential divisors is explicitly given. It is important to note that very few examples of this type are found in the literature.

math.AG↗