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Michel Brion

Publications and source records attributed to Michel Brion.

At least 19 recordsLinked to original sources
arXiv2025

Minimal rational curves on complete symmetric varieties

We describe the families of minimal rational curves on any complete symmetric variety, and the corresponding varieties of minimal rational tangents (VMRT). In particular, we prove that these varieties are homogeneous and that for non-exceptional irreducible wonderful varieties, there is a unique family of minimal rational curves, and hence a unique VMRT. We relate these results to the restricted root system of the associated symmetric space. In particular we answer by the negative a question of Hwang: for certain Fano wonderful symmetric varieties, the VMRT has two connected components.

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arXiv2024

The inverse Galois problem for connected algebraic groups

We show that each connected group scheme of finite type over an arbitrary ground field is isomorphic to the component of the identity inside the automorphism group scheme of some projective, geometrically integral scheme. The main ingredients are embeddings into smooth group schemes, equivariant completions, blow-ups of orbit closures, Fitting ideals for Kähler differentials, and Blanchard's Lemma.

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arXiv2024

Actions of finite group schemes on curves

Every action of a finite group scheme $G$ on a variety admits a projective equivariant model, but not necessarily a normal one. As a remedy, we introduce and explore the notion of $G$-normalization. In particular, every curve equipped with a $G$-action has a unique projective $G$-normal model, characterized by the invertibility of ideal sheaves of all orbits. Also, $G$-normal curves occur naturally in some questions on surfaces in positive characteristics.

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arXiv2024

Equivariantly normal varieties for diagonalizable group actions

Given a finite group scheme $G$ over a field and a $G$-variety $X$, we obtain a criterion for $X$ to be $G$-normal in the sense of \cite{Br24}. When $G$ is diagonalizable, we describe the local structure of $G$-normal varieties in codimension $1$ and their dualizing sheaf. As an application, we obtain a version of the Hurwitz formula for $G$-normal varieties, where $G$ is linearly reductive.

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arXiv2022

On models of algebraic group actions

We show that every action of a smooth algebraic group on a variety admits a normal projective model. Along the way, we present new proofs of some basic results on algebraic transformation groups, including Weil's regularization theorem.

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arXiv2021

Homomorphisms of algebraic groups: representability and rigidity

Given two algebraic groups $G$, $H$ over a field $k$, we investigate the representability of the functor of morphisms (of schemes) $\mathbf{Hom}(G,H)$ and the subfunctor of homomorphisms (of algebraic groups) $\mathbf{Hom}_{\rm gp}(G,H)$. We show that $\mathbf{Hom}(G,H)$ is represented by a group scheme, locally of finite type, if the $k$-vector space $\mathcal{O}(G)$ is finite-dimensional; the converse holds if $H$ is not étale. When $G$ is linearly reductive and $H$ is smooth, we show that $\mathbf{Hom}_{\rm gp}(G,H)$ is represented by a smooth scheme $M$; moreover, every orbit of $H$ acting by conjugation on $M$ is open.

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arXiv2020

Minimal rational curves on generalized Bott-Samelson varieties

We investigate families of minimal rational curves on Schubert varieties, their Bott-Samelson desingularizations, and their generalizations constructed by Nicolas Perrin in the minuscule case. In particular, we describe the minimal families on small resolutions of minuscule Schubert varieties.

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arXiv2020

Some automorphism groups are linear algebraic

Consider a normal projective variety $X$, a linear algebraic subgroup $G$ of Aut($X$), and the field $K$ of $G$-invariant rational functions on $X$. We show that the subgroup of Aut($X$) that fixes $K$ pointwise is linear algebraic. If $K$ has transcendence degree $1$ over $k$, then Aut($X$) is an algebraic group.

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arXiv2020

Very special algebraic groups

We say that a smooth algebraic group $G$ over a field $k$ is very special if for any field extension $K/k$, every $G_K$-homogeneous $K$-variety has a $K$-rational point. It is known that every split solvable linear algebraic group is very special. In this note, we show that the converse holds, and discuss its relationship with the birational classification of algebraic group actions.

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arXiv2020

Homogeneous vector bundles over abelian varieties via representation theory

Let $A$ be an abelian variety over a field. The homogeneous (or translation-invariant) vector bundles over $A$ form an abelian category ${\rm HVec}_A$; the Fourier-Mukai transform yields an equivalence of ${\rm HVec}_A$ with the category of coherent sheaves with finite support on the dual abelian variety. In this paper, we develop an alternative approach to homogeneous vector bundles, based on the equivalence of ${\rm HVec}_A$ with the category of finite-dimensional representations of a commutative affine group scheme (the "affine fundamental group" of $A$). This displays remarkable analogies between homogeneous vector bundles over abelian varieties and representations of split reductive algebraic groups.

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arXiv2019

Automorphism groups of almost homogeneous varieties

Consider a smooth connected algebraic group $G$ acting on a normal projective variety $X$ with an open dense orbit. We show that Aut($X$) is a linear algebraic group if so is $G$; for an arbitrary $G$, the group of components of Aut($X$) is arithmetic. Along the way, we obtain a restrictive condition for $G$ to be the full automorphism group of some normal projective variety.

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arXiv2019

Some combinatorial aspects of generalised Bott-Samelson varieties

We obtain two combinatorial results: an equality of Weyl groups and an inequality of roots, in the setting of generalised Bott-Samelson resolutions of minuscule Schubert varieties. These results are used in the companion paper [BK19] to describe minimal rational curves on these resolutions, and their relation to lines on the Schubert varieties.

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arXiv2019

On the fundamental groups of commutative algebraic groups

Consider the abelian category ${\mathcal C}$ of commutative group schemes of finite type over a field $k$, its full subcategory ${\mathcal F}$ of finite group schemes, and the associated pro category ${\rm Pro}({\mathcal C})$ (resp. ${\rm Pro}({\mathcal F})$) of pro-algebraic (resp. profinite) group schemes. When $k$ is perfect, we show that the profinite fundamental group $\varpi_1 : {\rm Pro}({\mathcal C}) \to {\rm Pro}({\mathcal F})$ is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors $\varpi_i$ vanish for $i \geq 2$. Along the way, we describe the indecomposable projective objects of ${\rm Pro}({\mathcal C})$ over an arbitrary field $k$.

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arXiv2017

On actions of connected algebraic groups

We obtain a version of the theorem of the square and a local structure result for actions of connected algebraic groups on seminormal varieties in characteristic 0, and arbitrary varieties in positive characteristics.

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arXiv2017

Algebraic group actions on normal varieties

Let $G$ be a connected algebraic $k$-group acting on a normal $k$-variety, where $k$ is a field. We show that $X$ is covered by open $G$-stable quasi-projective subvarieties; moreover, any such subvariety admits an equivariant embedding into the projectivization of a $G$-linearized vector bundle on an abelian variety, quotient of $G$. This generalizes a classical result of Sumihiro for actions of smooth connected affine algebraic groups.

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