arXiv subjects

Giancarlo Lucchini Arteche

Publications and source records attributed to Giancarlo Lucchini Arteche.

At least 19 recordsLinked to original sources

arXiv2026

Equivariant automorphism group and real forms of complexity-one varieties

Let $G$ be a connected reductive real algebraic group. We prove that every real $G$-variety of complexity one admits only finitely many pairwise non-isomorphic $(\mathbb{R},G)$-forms. Our approach relies on representability and structural results for equivariant automorphism groups. More generally, over a perfect field, the equivariant automorphism group of an almost homogeneous variety under a smooth group scheme of finite type is represented by a smooth group scheme of finite type, and is linear whenever the acting group is linear. In characteristic zero, the equivariant automorphism group of every complexity-one variety under a connected reductive group is represented by a smooth group scheme locally of finite type. In the case that is not almost homogeneous, we further describe the subgroup acting trivially on the rational quotient as an extension of an étale group scheme, locally isomorphic to $\mathbb{Z}^m$, by a group of multiplicative type.

math.AG↗

arXiv2026

The structure of automorphism groups of zero-dimensional monomial algebras

Let $A$ be a zero-dimensional monomial algebra over an algebraically closed field of characteristic zero, that is, a finite-dimensional quotient of a polynomial ring by a monomial ideal. Its automorphism group $G$ is a linear algebraic group, described through the homogeneous nilpotent derivations of $A$. We analyze the structure of $G$ in detail. Its identity component $G^0$ is a semidirect product of its unipotent radical and a reductive subgroup isomorphic to a product of general linear groups, and for each root degree we characterize when the associated derivations give rise to an additive root subgroup, and determine its dimension. Using the Lie brackets of these derivations, we then give an explicit algorithm that produces, out of the minimal monomial generators of the ideal, a family of root subgroups generating $G^0$ together with a maximal torus. Such a family is minimal in the generic case. We also show that the component group $G/G^0$ can be arbitrary: every finite group arises as the component group of the automorphism group of some zero-dimensional monomial algebra. Finally, we apply these results to the algebras $\mathbf{k}[\mathbf{x}]/\mathfrak{m}^d$, showing that the subgroup generated by a maximal torus and the outer root subgroups is exactly the subgroup of automorphisms with constant Jacobian determinant, and we deduce from this a new proof of Anick's theorem on the density of the tame automorphisms of $\mathbf{k}[\mathbf{x}]$.

math.AC↗

arXiv2026

Finite-dimensional monomial algebras are determined by their automorphism group

A monomial algebra is the quotient of a polynomial algebra by an ideal generated by monomials. We prove that finite-dimensional monomial algebras are characterized by their automorphism group among finite-dimensional, local algebras with cotangent space of fixed dimension. In particular, we show how to recover a monomial ideal given the automorphism group of the corresponding monomial algebra.

math.AC↗

arXiv2026

Real approximation for homogeneous spaces with finite stabilizers

We prove some new cases of real appoximation for homogeneous spaces with finite stabilizers and describe the state of the art around this question, giving proofs that are well-known to experts but that, to our knowledge, cannot be found in the literature. Our main new result needs the latest advances in the topic of the Brauer--Manin obstruction for homogeneous spaces with supersolvable stabilizers. It states that any finite $k$-group that is split by a $2$-primary extension satisfies real approximation.

math.AG↗

arXiv2025

Transfer principles for Galois cohomology and Serre's conjecture II

In this article, we prove several transfer principles for the cohomological dimension of fields. Given a fixed field $K$ with finite cohomological dimension $δ$, the two main ones allow to: - construct totally ramified extensions of $K$ with cohomological dimension $\leq δ- 1$ when $K$ is a complete discrete valuation field; - construct algebraic extensions of $K$ with cohomological dimension $\leq δ-1$ and satisfying a norm condition. We then apply these results to Serre's conjecture II and to some variants for fields of any cohomological dimension that are inspired by conjectures of Kato and Kuzumaki. In particular, we prove that Serre's conjecture II for characteristic $0$ fields implies Serre's conjecture II for positive characteristic fields.

math.NT↗

arXiv2025

Central simple algebras, Milnor $K$-theory and homogeneous spaces over complete discretely valued fields of dimension 2

Let $K$ be a complete discretely valued field with residue field $\bar K$ of dimension $1$ (not necessarily perfect). This occurs if and only if $K$ has dimension $2$. We prove the following statements on the arithmetic of such fields: - The "period equals index" property holds for central simple $K$-algebras. - For every prime $p$, every class in the Milnor $\mathrm{K}$-theory modulo $p$ is represented by a symbol. - Serre's Conjecture II holds for the field $K$. That is, for every semisimple and simply connected $K$-group $G$, the set $H^1(K,G)$ is trivial.

math.RA↗

arXiv2023

Lifting vector bundles to Witt vector bundles

Let $X$ be a scheme. Let $r \geq 2$ be an integer. Denote by $W_r(X)$ the scheme of Witt vectors of length $r$, built out of $X$. We are concerned with the question of extending (=lifting) vector bundles on $X$, to vector bundles on $W_r(X)$-promoting a systematic use of Witt modules and Witt vector bundles. To begin with, we investigate two elementary but significant cases, in which the answer to this question is positive: line bundles, and the tautological vector bundle of a projective bundle over an affine base. We then offer a simple (re)formulation of classical results in deformation theory of smooth varieties over a field $k$ of characteristic $p>0$, and extend them to reduced $k$-schemes. Some of these results were recently recovered, in another form, by Stefan Schröer. As an application, we prove that the tautological vector bundle of the Grassmannian $Gr_{\mathbb{F}_p}(m,n)$ does not extend to $W_2(Gr_{\mathbb{F}_p}(m,n))$, if $2 \leq m \leq n-2$. To conclude, we establish a connection to the work of Zdanowicz, on non-liftability of some projective bundles.

math.AG↗

arXiv2023

On Kato and Kuzumaki's properties for the Milnor $K_2$ of function fields of $p$-adic curves

Let $K$ be the function field of a curve $C$ over a $p$-adic field $k$. We prove that, for each $n, d \geq 1$ and for each hypersurface $Z$ in $\mathbb{P}^n_{K}$ of degree $d$ with $d^2 \leq n$, the second Milnor $K$-theory group of $K$ is spanned by the images of the norms coming from finite extensions $L$ of $K$ over which $Z$ has a rational point. When the curve $C$ has a point in the maximal unramified extension of $k$, we generalize this result to hypersurfaces $Z$ in $\mathbb{P}^n_{K}$ of degree $d$ with $d \leq n$.

math.AG↗

arXiv2023

On composition of torsors

Let $K$ be a field, let $X$ be a connected smooth $K$-scheme and let $G,H$ be two smooth connected $K$-group schemes. Given $Y \to X$ a $G$-torsor and $Z \to Y$ an $H$-torsor, we study whether one can find an extension $E$ of $G$ by $H$ so that the composite $Z \to X$ is an $E$-torsor. We give both positive and negative results, depending on the nature of the groups $G$ and $H$.

math.AG↗

arXiv2021

Quotients of the Bruhat-Tits tree by arithmetic subgroups of special unitary groups

Let $K$ be the function field of a curve $C$ over a field $\mathbb{F}$ of either odd or zero characteristic. Following the work by Serre and Mason on $\mathrm{SL}_2$, we study the action of arithmetic subgroups of $\mathrm{SU}(3)$ on its corresponding Bruhat-Tits tree associated to a suitable completion of $K$. More precisely, we prove that the quotient graph "looks like a spider", in the sense that it is the union of a set of cuspidal rays (the "legs"), parametrized by an explicit Picard group, that are attached to a connected graph (the "body"). We use this description in order to describe these arithmetic subgroups as amalgamated products and study their homology. In the case where $\mathbb{F}$ is a finite field, we use a result by Bux, Köhl and Witzel in order to prove that the "body" is a finite graph, which allows us to get even more precise applications.

math.GR↗

arXiv2021

Local-global principles for homogeneous spaces over some two-dimensional geometric global fields

In this article, we study the obstructions to the local-global principle for homogeneous spaces with connected or abelian stabilizers over finite extensions of the field $\mathbb{C}((x,y))$ of Laurent series in two variables over the complex numbers and over function fields of curves over $\mathbb{C}((t))$. We give examples that prove that the usual Brauer-Manin obstruction is not enough to explain the failure of the local-global principle, and we then construct a variant of this obstruction using torsors under quasi-trivial tori which turns out to work. In the end of the article, we compare this new obstruction to the descent obstruction with respect to torsors under tori. For that purpose, we use a result on towers of torsors, that is of independent interest and therefore is proved in a separate appendix.

math.AG↗

arXiv2021

Smooth quotients of abelian surfaces by finite groups that fix the origin

Let $A$ be an abelian surface and let $G$ be a finite group of automorphisms of $A$ fixing the origin. Assume that the analytic representation of $G$ is irreducible. We give a classification of the pairs $(A,G)$ such that the quotient $A/G$ is smooth. In particular, we prove that $A=E^2$ with $E$ an elliptic curve and that $A/G\simeq\mathbb{P}^2$ in all cases. Moreover, for fixed $E$, there are only finitely many pairs $(E^2,G)$ up to isomorphism. This fills a small gap in the literature and completes the classification of smooth quotients of abelian varieties by finite groups fixing the origin started by the first two authors.

math.AG↗

arXiv2021

Smooth quotients of principally polarized abelian varieties

We give an explicit characterization of all principally polarized abelian varieties $(A,Θ)$ such that there is a finite subgroup of automorphisms $G$ of $A$ that preserve the numerical class of $Θ$, and such that the quotient variety $A/G$ is smooth. We also give a complete classification of smooth quotients of Jacobians of curves.

math.AG↗

arXiv2021

Smooth quotients of complex tori by finite groups (with an appendix by Stephen Griffeth)

Let $A$ be a complex torus and $G$ a finite group acting on $A$ without translations such that $A/G$ is smooth. Consider the subgroup $F\leq G$ generated by elements that have at least one fixed point. We prove that there exists a point $x\in A$ fixed by the whole group $F$ and that the quotient $A/G$ is a fibration of products of projective spaces over an étale quotient of a complex torus (the étale quotient being Galois with group $G/F$). In particular, when $G=F$, we may assume that $G$ fixes the origin. This is related to previous work by the authors, where the case of actions on abelian varieties fixing the origin was treated. Here, we generalize these results to complex tori and use them to reduce the problem of classifying smooth quotients of complex tori to the case of étale quotients. An ingredient of the proof of our fixed-point theorem is a result proving that in every irreducible complex reflection group there is an element which is not contained in any proper reflection subgroup and that Coxeter elements have this property for well-generated groups. This result is proved by Stephen Griffeth in an appendix.

math.AG↗

arXiv2021

On homogeneous spaces with finite anti-solvable stabilizers

We say that a group is anti-solvable if all of its composition factors are non-abelian. We consider a particular family of anti-solvable finite groups containing the simple alternating groups for $n\neq 6$ and all 26 sporadic simple groups. We prove that, if $K$ is a perfect field and $X$ is a homogeneous space of a smooth algebraic $K$-group $G$ with finite geometric stabilizers lying in this family, then $X$ is dominated by a $G$-torsor. In particular, if $G=\mathrm{SL}_n$, all such homogeneous spaces have rational points.

math.AG↗

arXiv2021

Automorphisms of products of toric varieties

We give an explicit description of the automorphism group of a product of complete toric varieties over an arbitrary field in terms of the respective automorphism groups of its components. More precisely, we prove that, up to permutation of isomorphic components, an automorphism of a product corresponds to a product of automorphisms of its components. We also reprove, in modern language, the classic result by Demazure describing the group-scheme of automorphisms of a complete toric variety over an arbitrary field.

math.AG↗

arXiv2020

A decomposition of the Jacobian of a Humbert-Edge curve

A \textit{Humbert-Edge curve of type} $n$ is a non-degenerate smooth complete intersection of $n-1$ diagonal quadrics. Such a curve has an interesting geometry since it has a natural action of the group $(\mathbb{Z}/2\mathbb{Z})^n$. We present here a decomposition of its Jacobian variety as a product of Prym-Tyurin varieties, and we compute the kernel of the corresponding isogeny.

math.AG↗

arXiv2020

On extensions of algebraic groups

We extend to the context of algebraic groups a classic result on extensions of abstract groups relating the set of isomorphism classes of extensions of $G$ by $H$ with that of extensions of $G$ by the center $Z$ of $H$. The proof should be easily generalizable to other contexts. We also study the subset of classes of split extensions and give a quick application by proving a finiteness result on these sets over a finite field.

math.AG↗