arXiv subjects

Giulia Battiston

Publications and source records attributed to Giulia Battiston.

8 recordsLinked to original sources

arXiv2018

Representations of Affine Group Schemes Over General Rings

Among all affine, flat, finitely presented group schemes, we focus on those that are pure, this includes all groups which are extensions of a finite locally free group by a group with connected fibres. We prove that over an arbitrary base ring, pure group schemes have a classifying space satisfying the resolution property, an embedding into some GLn, a tensor generator for their category of finite type representations, and can be reconstructed from their category of projective finite type representations. In the case of an Artinian base ring, the same is true for all affine, flat, finitely presented group schemes, this answers a question of Conrad. We also prove that quotients of pure groups by closed pure subgroups over an arbitrary base scheme are Zariski-locally quasi-projective. This answers a question of Raynaud, in the case of affine groups. We give various applications.

math.AG↗

arXiv2018

The normal hull and commutator group for nonconnected group schemes

In this short note, we prove that there is a well behaved notion of normal hull for smooth algebraic group schemes over a field and that the commutator group $(G,H)$ is well defined for $H\subset G$ smooth, even when both of them are not connected.

math.AG↗

arXiv2016

Gieseker conjecture for homogeneous spaces

We prove Gieseker conjecture for an homogeneous space $X$, saying that if $X$ has no non-trivial tame coverings then it has no non-trivial regular singular $\mathscr{O}_X$-coherent $\mathscr{D}_{X/k}$-modules. In order to do so we prove a Künneth formula for the regular singular stratified fundamental group and a base change for Gauss-Manin stratifications in the non-proper case.

math.AG↗

arXiv2016

The homotopy sequence for regular singular stratified bundles

A separable, proper morphism of varieties with geometrically connected fibers induces a homotopy exact sequence relating the étale fundamental groups of source, target and fiber. Extending work of dos Santos, we prove the existence of an analogous homotopy exact sequence for fundamental group schemes classifying regular singular stratified bundles, under the additional assumption that the morphism in question can be (partially) compactified to a log smooth morphism.

math.AG↗

arXiv2016

The base change of the monodromy group for geometric Tannakian pairs

We prove that in any characteristic the formation of the monodromy group of a $\mathscr{D}$-module commutes with the extension of the ground field, extending a result of Gabber for separable extensions.

math.AG↗

arXiv2015

The Casas-Alvero conjecture

We present a proof of the Casas-Alvero conjecture, stating that if a complex polynomial has a root in common with each of its derivatives it must be a multiple of the power of some monomial.

math.AG↗

arXiv2015

A theory of Galois descent for finite inseparable extensions

We present a generalization of Galois descent to finite modular normal field extension $L/K$, using the Heerma-Galois group $Aut(L[\bar{X}]/K[\bar{X}])$ where $L[\bar{X}]=L[X]/(X^{p^e})$ and $e$ is the exponent of $L$ over $K$.

math.AG↗

arXiv2015

The variation of the monodromy group in families of stratified bundles in positive characteristic

In this article we study smooth families of stratified bundles in positive characteristic and the variation of their monodromy group.Our aim is, in particular, to strengthen the weak form of the positive equicharacteristic $p$-curvature conjecture stated and proved by Esnault and Langer in "On a positive equicharacteristic variant of the $p$-curvature conjecture" (Doc. Math. 18 (2013)). The main result is that if the ground field is uncountable then the strong form holds. In the case where the ground field is countable we provide positive and negative answers to possible generalizations.

math.AG↗