arXiv subjects

Gilberto B. Almeida Filho

Publications and source records attributed to Gilberto B. Almeida Filho.

6 recordsLinked to original sources

arXiv2026

$a$-numbers of covers in characteristic two: semigroups and Riemann-Roch spaces

We compute the $a$-numbers of the Garcia-Güneri-Stichtenoth curves, the Beelen-Montanucci curves, and the Skabelund covers of Suzuki curves in characteristic two. The proofs use the identity $a(X)=\ell(\tfrac12\operatorname{div}(dt))$ and reduce the calculations to Riemann-Roch spaces on the base curves. We obtain transfer formulas for odd-degree radical covers, with one or several points at infinity, and evaluate them using Weierstrass semigroups and plane sections of the Hermitian curve. For the first two families we also give cyclic models over the projective line and relate the formulas to Elkin's rank formula. We then treat two classes of Artin-Schreier fibre products. A supersingular genus-six family has a-number $3$ at the zero parameter and $2$ elsewhere, although the genera and $a$-numbers of its quadratic quotients remain fixed. We determine its Ekedahl--Oort types as $(0,0,0,1,2,3)$ and $(0,1,1,2,3,4)$,

math.AG↗

arXiv2026

$a$-Numbers, higher Cartier ranks and maximality of van der Geer-van der Vlugt curves

Let $p$ be an odd prime and let $C_R$ be the van der Geer-van der Vlugt curve $y^p-y=xR(x)$ over an algebraically closed field of characteristic $p$, where $R(x)=\sum_{i=0}^{h}a_ix^{p^i}$ is an additive polynomial of degree $p^h$. We determine the image and the rank of every power of the Cartier operator $\Cart$ on $H^0(C_R,\Omg)$. For $1\le r\le h-1$ the image of $\Cart^r$ is spanned by an explicit set of monomial differentials that does not depend on $R$, and $\rank(\Cart^r)=p^{h-r-1}\binom{p+r}{r+2}$. The last iterate detects the linear coefficient: $\rank(\Cart^h)=0$ if $a_0=0$ and $\rank(\Cart^h)=\binom{(p-1)/2+h}{h+1}$ if $a_0\neq0$, so the nilpotency index of $\Cart$ is $h$ or $h+1$. Hence, for $h\ge2$, the loci $a_0=0$ and $a_0\ne0$ have the same $a$-number, which is the largest $a$-number of a one-point Artin-Schreier curve with pole order $p^h+1$, but they share no Ekedahl-Oort type. Over finite fields we deduce that a curve over $\F_{p^{2n}}$ that is geometrically isomorphic to $C_R$ can be $\F_{p^{2n}}$-maximal or $\F_{p^{2n}}$-minimal only if $n\ge h$, and only if $n\ge h+1$ when $a_0\neq0$. Explicit examples show that the second bound is attained for $(p,h)=(3,1)$, $(3,2)$ and $(5,2)$.

math.AG↗

arXiv2026

Nonexistence of maximal curves of genus five over $\F_{64}$

We show that there is no maximal curve of genus five over $\F_{64}$. As a consequence, $N_{64}(5)=140$, and the genus spectrum of maximal curves over $\F_{64}$ is determined. The proof uses the vanishing of the third iterate of the Cartier operator. We prove that a nonhyperelliptic curve of genus five in characteristic two satisfying this condition is nontrigonal. Its canonical theta characteristic defines a separable cover of degree four with one geometric branch value. The two possible ramification types give either a rational subcanonical point or a point bound obtained from the cubic resolvent. Both cases exclude maximality over $\F_{64}$.

math.AG↗

arXiv2026

Maximal Subcovers of the Skabelund Curve: Uniqueness via Genus and Automorphism Groups

We establish a rigidity phenomenon for a family of intermediate covers of the Skabelund curve over $\mathbb{F}_{q^4}$. The Skabelund curve, introduced by D.~Skabelund as a cyclic cover of the Suzuki curve, is a maximal curve with a large automorphism group and plays a central role in the theory of maximal curves over finite fields. For the intermediate covers arising from this construction, we determine their full automorphism groups and compute the Weierstrass semigroups at all $\mathbb{F}_{q^4}$-rational points. Using these structural and arithmetic invariants, we prove that each curve in the family is uniquely determined, up to isomorphism over its field of definition, by the pair consisting of its genus and its full automorphism group. This provides a rigidity-type classification of intermediate Suzuki-type covers; in particular, the Skabelund curve itself is uniquely characterized within this family by its genus and automorphism group.

math.AG↗

arXiv2022

Gapsets and the $k$-generalized Fibonacci sequences

In this paper, we bring the terminology of the Kunz coordinates of numerical semigroups to gapsets and we generalize this concept to $m$-extensions. It allows us to identify gapsets and, in general, $m$-extensions with tilings of boards. As a consequence, we prove a version of Bras-Amorós conjecture for $m$-extensions. Besides, we obtain a lower bound for the number of gapsets with fixed genus and depth at most 3 and a family of upper bounds for the number of gapsets with fixed genus. Moreover, we present explicit formulas for the number of gapsets with fixed genus and depth, when the multiplicity is 3 or 4, and, in some cases, for the number of gapsets with fixed genus and depth.

math.CO↗

arXiv2022

On Pure k-sparse gapsets

In this paper, we study gapsets and we focus on obtaining information on how the maximum distance between to consecutive elements influences the behaviour of the set. In particular, we prove that the cardinality of the set of gapsets with genus $g$ such that the maximum distance between two consecutive elements is $κ$ is equal to the cardinality of the set of gapsets with genus $g+1$ such that the maximum distance between two consecutive elements is $κ+1$, when $2g \leq 3κ$.

math.CO↗