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Alvaro Rittatore

Publications and source records attributed to Alvaro Rittatore.

13 recordsLinked to original sources
arXiv2024

On polynomial automorphisms commuting with a simple derivation

Let $D$ be a simple derivation of the polynomial ring $\mathbb{k}[x_1,\dots,x_n]$, where $\mathbb{k}$ is an algebraically closed field of characteristic zero, and denote by $\operatorname{Aut}(D)\subset\operatorname{Aut}(\mathbb{k}[x_1,\dots,x_n])$ the subgroup of $\mathbb{k}$-automorphisms commuting with $D$. We show that the connected component of $\operatorname{Aut}(D)$ passing through the identity is a unipotent algebraic group of dimension at most $n-2$, this bound being sharp. Moreover, $\operatorname{Aut}(D)$ is an algebraic group if and only if it is a connected ind-group. Given a simple derivation $D$, we characterize when $\operatorname{Aut}(D)$ contains a normal subgroup of translations. As an application of our techniques we show that if $n=3$, then either $\operatorname{Aut}(D)$ is a discrete group or it is isomorphic to the additive group acting by translations, and give some insight on the case $n=4$.

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arXiv2023

Gorenstein Fano Generic Torus Orbit closures in $G/P$

Given a reductive group $G$ and a parabolic subgroup $P\subset G$, with maximaltorus $T$, we consider (following Dabrowski's work) the closure $X$ of a generic $T$-orbit in $G/P$, and determine in combinatorial termswhen the toric variety $X$ is $\mathbb{Q}$-Gorenstein Fano, extending in this way the classification of smooth Fano generic closures given by Voskresenski\uı and Klyachko. As an application, we apply the well known correspondence between Gorenstein Fano toric varieties and reflexive polytopes in order to exhibit which reflexive polytopes correspond to generic closures -- this list includes the reflexive root polytopes.

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arXiv2022

On the orbits of plane automorphisms and their stabilizers

Let $\Bbbk$ be a perfect field with algebraic closure $\overline{\Bbbk}$. If $H$ is a subgroup of plane automorphisms over $\Bbbk$ and $p\in\overline{\Bbbk}^2$ is a point, we describe the subgroup consisting of plane automorphisms which stabilize the orbit of $p$ under $H$, when this orbit has irreducible closure in $\overline{\Bbbk}^2$. As an application, we treat the case where $H$ is cyclic and the closure of the orbit of $p$ is an arbitrary (non-necessarily irreducible) curve.

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arXiv2020

Quasi-compact group schemes, Hopf sheaves, and their representations

We explore the notion of representation of an affine extension of an abelian variety -- such an extension is a faithfully flat affine morphism of $\Bbbk$-group schemes $q:G\to A$, where $A$ is an abelian variety. We characterize the categories that arise as the category of representations of an affine extension $q:G\to A$, generalizing the classical results of Tannaka Duality established for affine $\Bbbk$-group schemes (that is, when $A=\operatorname{Spec}(\Bbbk)$). We also prove the existence of a contravariant equivalence between the category of affine extensions of a given $A$ and the category of faithful commutative Hopf sheaves on $A$, generalizing in this manner the well-known op-equivalence between affine group schemes and commutative Hopf algebras. If $\mathcal H_q$ is the Hopf sheaf on $A$ associated to $q$, the category of representations of $q$ is equivalent to the category of $\mathcal H_q$-comodules.

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arXiv2013

Some remarks about the Zariski topology of the Cremona group

For an algebraic variety $X$ we study the behavior of algebraic morphisms from an algebraic variety to the group $\bir(X)$ of birational maps of $X$ and obtain, as application, some insight about the relationship between the so-called Zariski topology of $\bir(X)$ and the algebraic structure of this group, where $X$ is a rational variety.

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arXiv2011

Self-dual projective toric varieties

Let T be a torus over an algebraically closed field k of characteristic 0, and consider a projective T-module P(V). We determine when a projective toric subvariety X of P(V) is self-dual, in terms of the configuration of weights of V.

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arXiv2009

Observable subgroups of algebraic monoids

A closed subgroup H of the affine, algebraic group G is called observable if G/H is a quasi-affine algebraic variety. In this paper we define the notion of an observable subgroup of the affine, algebraic monoid M. We prove that a subgroup H of G is observable in M if and only if H is closed in M and there are "enough" H-semiinvariant functions in K[M]. We show also that a closed, normal subgroup H of G (the unit group of M) is observable in M if and only if it is closed in M. In such a case there exists a determinant $χ: M \to K$ such that $H\subset ker(χ)$. As an application, we show that in this case the affinized quotient $M/_{aff} H$ of M by H is an affine algebraic monoid scheme with unit group G/H.

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arXiv2009

The ring of regular functions of an algebraic monoid

Let M be an irreducible normal algebraic monoid with unit group G. It is known that G admits a Rosenlicht decomposition, G=G_antG_aff, where G_ant is the maximal anti-affine subgroup of G, and G_aff the maximal normal connected affine subgroup of G. In this paper we show that this decomposition extends to a decomposition M=G_antM_aff, where M_aff is the affine submonoid M_aff=\bar{G_aff}. We then use this decomposition to calculate $\mathcal{O}(M)$ in terms of $\mathcal{O}(M_aff)$ and G_aff, G_ant\subset G. In particular, we determine when M is an anti-affine monoid, that is when $\mathcal{O}(M)=K$.

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arXiv2009

Observable actions of algebraic groups

Let G be an affine algebraic group and let X be an affine algebraic variety. An action $G\times X \to X$ is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant $f\in K[X]^G$ such that f(Y) =0. We characterize this condition geometrically as follows. The action $G\times X \to X$ is observable if and only if (1) there is a nonempty open subset $U\subseteq X$ consisting of closed orbits, and (2) the field $K(X)^G$ of G-invariant rational functions on X is equal to the quotient field of $K[X]^G$. In case G is reductive, we conclude that there exists a unique, maximal, G-stable, closed subset $X_{\soc}$ of $X$ such that $G\times X_{\soc} \to X_{\soc}$ is observable. Furthermore, the canonical map $X_{\soc}// G \to X//G$ is finite and bijective.

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arXiv2007

The structure of normal algebraic monoids

We show that any normal algebraic monoid is an extension of an abelian variety by a normal affine algebraic monoid. This extends (and builds on) Chevalley's structure theorem for algebraic groups.

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arXiv2005

Generalized Cayley's $Ω$-processes

In this paper we generalize some constructions and results due to Cayley and Hilbert. We define the concept of $Ω$--process for an arbitrary algebraic monoid with zero and unit group $G$. Then we show how to produce from the process and for a linear rational representation of $G$, a number of elements of the ring of $G$-invariants, that is large enough as to guarantee its finite generation. Moreover, we give an explicit construction of all $Ω$-processes for general reductive monoids and, in the case of the monoid of all the $n^2$ matrices, compare our construction with Cayley's definition.

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