arXiv subjects

Bianca Gouthier

Publications and source records attributed to Bianca Gouthier.

5 recordsLinked to original sources

arXiv2026

Infinitesimal commutative unipotent group schemes with one-dimensional Lie algebra

We prove that over an algebraically closed field of characteristic $p>0$ there are exactly, up to isomorphism, $n$ infinitesimal commutative unipotent $k$-group schemes of order $p^n$ with one-dimensional Lie algebra, and we explicitly describe them. We consequently obtain an explicit description of all infinitesimal subgroup schemes of any supersingular elliptic curve over an algebraically closed field, recovering all their $p^n$-torsions as well. Finally, we use these results to answer a question of Brion on rational actions of infinitesimal commutative unipotent group schemes on curves.

math.AG↗

arXiv2026

Infinitesimal rational actions

For any finite $k$-group scheme $G$ acting rationally on a $k$-variety, if the action is generically free then the dimension of $\mathrm{Lie} (G)$ is upper bounded by the dimension of the variety. We show that this is the only obstruction when $k$ is a perfect field of positive characteristic and $G$ is infinitesimal commutative trigonalizable. We also give necessary conditions to have faithful rational actions of infinitesimal commutative trigonalizable group schemes on varieties, and (different) sufficient conditions in the unipotent case over a perfect field.

math.AG↗

arXiv2026

Weierstrass semigroups and the order bound

The aim of this survey is to provide the reader with an essential and accessible introduction to the theory of Weierstrass semigroups, in the context of the theory developed by K.-O. Stöhr and J.F. Voloch. Furthermore, we discuss an application of Stöhr-Voloch theory in coding theory, namely the Feng-Rao bound (also known as the order bound) for the dual minimum distance of one-point algebraic geometry codes from a curve, which relies on the knowledge of certain Weierstrass semigroups of the curve.

math.AG↗

arXiv2024

Unexpected subgroup schemes of PGL_{2,k} in characteristic 2

If the characteristic of a field $k$ is odd any infinitesimal group scheme of $PGL_{2,k}$ lifts to $SL_{2,k}$. In this paper, we prove that this is not true in characteristic $2$ and we give a complete description, up to isomorphism, of infinitesimal unipotent subgroup schemes of $PGL_{2,k}$. Also, the infinitesimal trigonalizable case is considered.

math.AG↗

arXiv2022

On the existence of "Spot It!" decks that are not projective planes

The game "Spot It!" is played with a deck of cards in which every pair of cards has exactly one matching symbol and the aim is to be the fastest at finding the match. It is known that finite projective planes correspond to decks in which every card contains $n$ symbols and every symbol appears on $n$ cards. In this paper we relax the hypothesis on the number of cards on which a symbol appears: we study symmetric decks in which every symbol appears the same number of times and we introduce the concept of maximal deck, providing a sufficient condition for a deck to have this property. We also produce various examples of interesting decks which do not correspond to projective planes.

math.CO↗