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Cesar Hilario

Publications and source records attributed to Cesar Hilario.

6 recordsLinked to original sources
arXiv2025

A class of geometrically elliptic fibrations by plane projective quartic curves

We investigate fibrations by non-hyperelliptic curves of arithmetic genus three and geometric genus one in characteristic two. Assuming that there is only one moving singularity and that its image in the Frobenius pullback of the fibration has degree one over the base, we provide a complete classification up to birational equivalence. This relies on an in-depth analysis of the generic fibres, whose geometry we describe explicitly. We prove that these fibrations are covered by elliptic fibrations, and that the covers are birational on the fibres but purely inseparable of exponent one on the bases.

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arXiv2024

On regular but non-smooth integral curves

Let $C$ be a regular geometrically integral curve over an imperfect field $K$ and assume that it admits a non-smooth point $\mathfrak{p}$ which -- seen as a prime of the separable function field $K(C)|K$ -- is non-decomposed in the base field extension $\overline{K} \otimes_K K(C)|\overline{K}$. In this paper we establish a bound for the number of iterated Frobenius pullbacks needed in order to transform $\mathfrak{p}$ into a rational point. This provides an algorithm to compute geometric $δ$-invariants of non-smooth points and a procedure to construct fibrations with moving singularities of prescribed $δ$-invariants. We show that the bound is sharp in characteristic 2. We further study the geometry of a pencil of plane projective rational quartics in characteristic 2 whose generic fibre attains our bound. On our way, we prove several results on separable and non-decomposed points that might be of independent interest.

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arXiv2024

Turning non-smooth points into rational points

In a recent paper, the author and St\"ohr established a bound on the number of iterated Frobenius pullbacks needed to transform a non-smooth purely inseparable point on a regular geometrically integral curve into a rational point. In this paper we improve this result, by establishing a new bound that is sharp in every characteristic $p>0$.

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arXiv2023

Fibrations by plane quartic curves with a canonical moving singularity

We classify fibrations by integral plane projective rational quartic curves whose generic fibre is regular but admits a non-smooth point that is a canonical divisor. These fibrations can only exist in characteristic two. The geometric generic fibre, which determines the generic behaviour of the special fibres, is an integral plane projective rational quartic curve over the algebraic closure of the function field of the base. It has the remarkable property that the tangent lines at the non-singular points are either all bitangents or all non-ordinary inflection tangents; moreover it is strange, that is, all the tangent lines meet in a common point. We construct two fibrations that are universal in the sense that any other fibration with the aforementioned properties can be obtained from one of them by a base extension. Furthermore, among these fibrations we choose a pencil of plane quartic curves and study in detail its geometry. We determine the corresponding minimal regular model and we describe it as a purely inseparable double covering of a quasi-elliptic fibration.

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arXiv2023

Generalizations of quasielliptic curves

We generalize the notion of quasielliptic curves, which have infinitesimal symmetries and exist only in characteristic two and three, to a remarkable hierarchy of regular curves having infinitesimal symmetries, defined in all characteristics and having higher genera. This relies on the study of certain infinitesimal group schemes acting on the affine line and certain compactifications. The group schemes are defined in terms of invertible additive polynomials over rings with nilpotent elements, and the compactification is constructed with the theory of numerical semigroups. The existence of regular twisted forms relies on Brion's recent theory of equivariant normalization. Furthermore, extending results of Serre from the realm of group cohomology, we describe non-abelian cohomology for semidirect products, to compute in special cases the collection of all twisted forms.

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