arXiv subjects
Publications and source records attributed to Thomas Bouchet.
In this article we give explicit equations for two Shimura families of genus 4 curves whose Jacobians are abelian fourfolds of Mumford type. These are the first explicit examples of abelian varieties over $\mathbb{Q}$ with endomorphism algebra $\mathbb{Z}$ and additional Hodge tensors.
Let $\mathcal{H}_g$ denote the moduli space of smooth hyperelliptic curves of genus $g$ in characteristic $p\geq 3$, and let $\mathcal{H}_g^f$ denote the $p$-rank $f$ stratum of $\mathcal{H}_g$ for $0 \leq f \leq g$. Achter and Pries note in their 2011 work that determining the number of irreducible components of $\mathcal{H}_g^f$ would lead to several intriguing corollaries. In this paper, we present a computational approach for estimating the number of irreducible components in various $p$-rank strata. Our strategy involves sampling curves over finite fields and calculating their $p$-ranks. From the data gathered, we conjecture that the non-ordinary locus is geometrically irreducible for all genera $g> 1$. The data also leads us to conjecture that the moduli space $\mathcal{H}^{g-2}_g$ is irreducible and suggests that $\mathcal{H}^f_g$ is irreducible for all $1 \leq f \leq g$. We conclude with a brief discussion on $\mathcal{H}^0_g$.
Let $K$ be a field of characteristic $0$. We present an explicit algorithm that, given the invariants of a generic homogeneous polynomial $f$ under the linear action of $\mathrm{GL}_n$ or $\mathrm{SL}_n$, returns a polynomial differing from $f$ only by a linear change of variables with coefficients in a finite extension of $K$. Our approach uses the theory of covariants and the Veronese embeddings to characterize the linear equivalence class of a homogeneous polynomial through equations whose coefficients are invariants. As applications, we derive explicit formulas for reconstructing of a generic non-hyperelliptic curve of genus 4 from its invariants, as well as reconstructing generic non-hyperelliptic curves of genus 3 from their Dixmier-Ohno invariants. In both cases, the coefficients of the reconstructed curve lie in its field of moduli.
The present paper gives an explicit classification of the isomorphism classes of non-hyperelliptic genus 4 curves over an algebraically closed field of characteristic 0. A non-hyperelliptic genus 4 curve lies on a quadric in $\mathbb{P^3}$ of rank 3 or 4. In the case of rank 3, we give a set of 60 invariants which classify the isomorphism classes, and in the case of rank 4, we find 65 invariants. These invariants are defined by transvectants and can be efficiently computed on a given example.
We describe algorithms based on invariant theory to solve problems on the geometry of curves, mainly those of genus 2, 3 and 4. New theoretical results building on the first author's PhD thesis are also included.