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Ronan Terpereau

Publications and source records attributed to Ronan Terpereau.

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.

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arXiv2026

Horospherical subgroups in positive characteristic

We investigate horospherical homogeneous spaces--a class of spherical homogeneous spaces encompassing both flag varieties and algebraic tori--over algebraically closed fields of characteristic p>0, and establish their complete classification for p>2.

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arXiv2026

Classification of equivariantly normal curves via Altmann-Hausen-Süss theory

Let the ground field be perfect of positive characteristic. Using Altmann-Hausen-Süss theory, we obtain a combinatorial classification of equivariantly normal curves with prescribed quotient in both the affine and projective settings. As a consequence, we derive an explicit upper bound on the number of isomorphism classes of equivariantly normal curves over a fixed base curve with prescribed branch locus. Furthermore, assuming that the ground field is algebraically closed, we determine, for a fixed cardinality of the branch locus, precisely when the set of isomorphism classes of equivariantly normal projective curves with prescribed quotient is finite and when it is infinite.

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arXiv2024

Real forms of Mori fiber spaces with many symmetries

We determine the rational real forms of the complex Mori fiber spaces for which the identity component of the automorphism group is a maximal connected algebraic subgroup of $\mathrm{Bir}(\mathbb{P}_{\mathbb{C}}^{3})$. This yields a list of maximal connected algebraic subgroup of $\mathrm{Bir}(\mathbb{P}_{\mathbb{R}}^{3})$. We furthermore determine the equivariant Sarkisov links starting from these rational real forms. This article is the first step towards classifying all the maximal connected algebraic subgroups of $\mathrm{Bir}(\mathbb{P}_{\mathbb{R}}^{3})$.

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arXiv2024

Forms of almost homogeneous varieties over perfect fields

We study the k-forms of almost homogeneous varieties over perfect base fields k. First, we discuss criteria for the existence of k-forms in the homogeneous case. Then, we extend the Luna-Vust theory from algebraically closed fields to perfect fields to determine when a given k-form of the open orbit of an almost homogeneous variety extends to a k-form of the entire variety. Finally, in the last section, we apply these results to determine the real forms of complex almost homogeneous SL(2)-threefolds.

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arXiv2023

Automorphisms of $\mathbb{P}^1$-bundles over rational surfaces

In this paper we provide the complete classification of $\mathbb{P}^1$-bundles over smooth projective rational surfaces whose neutral component of the automorphism group is maximal. Our results hold over any algebraically closed field of characteristic zero.

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arXiv2022

Real structures on nilpotent orbit closures

We determine the equivariant real structures on nilpotent orbits and the normalizations of their closures for the adjoint action of a complex semisimple algebraic group on its Lie algebra.

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arXiv2021

Connected algebraic groups acting on three-dimensional Mori fibrations

We study the connected algebraic groups acting on Mori fibrations $X \to Y$ with $X$ a rational threefold and $\mathrm{dim}(Y) \geq 1$. More precisely, for these fibre spaces we consider the neutral component of their automorphism groups and study their equivariant birational geometry. This is done using, inter alia, minimal model program and Sarkisov program and allows us to determine the maximal connected algebraic subgroups of $\mathrm{Bir}(\mathbb{P}^3)$, recovering most of the classification results of Hiroshi Umemura in the complex case.

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arXiv2021

Real structures on symmetric spaces

We obtain a necessary and sufficient condition for the existence of equivariant real structures on complex symmetric spaces for semisimple groups and discuss how to determine the number of equivalence classes for such structures.

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arXiv2021

Real structures on horospherical varieties

We study the equivariant real structures on complex horospherical varieties, generalizing classical results known for toric varieties and flag varieties. In particular, we obtain a necessary and sufficient condition for the existence of such real structures and determine the number of equivalence classes. We then apply our results to classify the equivariant real structures on smooth projective horospherical varieties of Picard rank 1.

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arXiv2019

Horospherical stacks

We prove structure theorems for algebraic stacks with a reductive group action and a dense open substack isomorphic to a horospherical homogeneous space, and thereby obtain new examples of algebraic stacks which are global quotient stacks. Our results partially generalize the work of Fantechi-Mann-Nironi and Geraschenko-Satriano for abstract toric stacks.

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arXiv2017

Stability conditions and related filtrations for $(G,h)$-constellations

Given an infinite reductive algebraic group $G$, we consider $G$-equivariant coherent sheaves with prescribed multiplicities, called $(G,h)$-constellations, for which two stability notions arise. The first one is analogous to the $θ$-stability defined for quiver representations by King and for $G$-constellations by Craw and Ishii, but depending on infinitely many parameters. The second one comes from Geometric Invariant Theory in the construction of a moduli space for $(G,h)$-constellations, and depends on some finite subset $D$ of the isomorphy classes of irreducible representations of $G$. We show that these two stability notions do not coincide, answering negatively a question raised in [BT15]. Also, we construct Harder-Narasimhan filtrations for $(G,h)$-constellations with respect to both stability notions (namely, the $μ_θ$-HN and $μ_D$-HN filtrations). Even though these filtrations do not coincide in general, we prove that they are strongly related: the $μ_θ$-HN filtration is a subfiltration of the $μ_D$-HN filtration, and the polygons of the $μ_D$-HN filtrations converge to the polygon of the $μ_θ$-HN filtration when $D$ grows.

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arXiv2017

Towards a symplectic version of the Chevalley restriction theorem

If $(G,V)$ is a polar representation with Cartan subspace $\mathfrak c$ and Weyl group $W$, it is shown that there is a natural morphism of Poisson schemes $\mathfrak c \oplus {\mathfrak c}^*/W \to V\oplus V^*/\!\!/\!\!/ G$. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if $(G,V)$ is visible. The conjecture is proved for visible stable locally free polar representations and certain further examples.

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arXiv2017

Moduli spaces of (G,h)-constellations

Given an infinite reductive group G acting on an affine scheme X over C and a Hilbert function h: Irr G \to N_0, we construct the moduli space M_θ(X) of θ-stable (G,h)-constellations on X, which is a generalization of the invariant Hilbert scheme after Alexeev and Brion and an analogue of the moduli space of θ-stable G-constellations for finite groups introduced by Craw and Ishii. Our construction of a morphism M_θ(X) \to X//G makes this moduli space a candidate for a resolution of singularities of the quotient X//G.

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arXiv2016

The Cox ring of a complexity-one horospherical variety

Cox rings are intrinsic objects naturally generalizing homogeneous coordinate rings of projective spaces. A complexity-one horospherical variety is a normal variety equipped with a reductive group action whose general orbit is horospherical and of codimension one. In this note, we provide a presentation by generators and relations for the Cox rings of complete rational complexity-one horospherical varieties.

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arXiv2015

On the geometry of normal horospherical G-varieties of complexity one

Let G be a connected simply-connected reductive algebraic group. In this article, we consider the normal algebraic varieties equipped with a horospherical G-action such that the quotient of a G-stable open subset is a curve. Let X be such a G-variety. Using the combinatorial description of Timashev, we describe the class group of X by generators and relations and we give a representative of the canonical class. Moreover, we obtain a smoothness criterion for X and a criterion to determine whether the singularities of X are rational or log-terminal respectively.

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arXiv2015

Invariant deformation theory of affine schemes with reductive group action

We develop an invariant deformation theory, in a form accessible to practice, for affine schemes $W$ equipped with an action of a reductive algebraic group $G$. Given the defining equations of a $G$-invariant subscheme $X \subset W$, we device an algorithm to compute the universal deformation of $X$ in terms of generators and relations up to a given order. In many situations, our algorithm even computes an algebraization of the universal deformation. As an application, we determine new families of examples of the invariant Hilbert scheme of Alexeev and Brion, where $G$ is a classical group acting on a classical representation, and describe their singularities.

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