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Gábor Korchmáros

Publications and source records attributed to Gábor Korchmáros.

25 records · Page 2Linked to original sources
arXiv2016

Ordinary algebraic curves with many automorphisms in positive characteristic

Let $\mathcal{X}$ be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus $\mathcal{g}(\mathcal{X}) \ge 2$ defined over an algebraically closed field $\mathbb{K}$ of odd characteristic $p$. Let $Aut(\mathcal{X})$ be the group of all automorphisms of $\mathcal{X}$ which fix $\mathbb{K}$ element-wise. For any solvable subgroup $G$ of $Aut(\mathcal{X})$ we prove that $|G|\leq 34 (\mathcal{g}(\mathcal{X})+1)^{3/2}$. There are known curves attaining this bound up to the constant $34$. For $p$ odd, our result improves the classical Nakajima bound $|G|\leq 84(\mathcal{g}(\mathcal{X})-1)\mathcal{g}(\mathcal{X})$, and, for solvable groups $G$, the Gunby-Smith-Yuan bound $|G|\leq 6(\mathcal{g}(\mathcal{X})^2+12\sqrt{21}\mathcal{g}(\mathcal{X})^{3/2})$ where $\mathcal{g}(\mathcal{X})>cp^2$ for some positive constant $c$.

math.AG↗
arXiv2016

3-nets realizing a diassociative loop in a projective plane

A \textit{$3$-net} of order $n$ is a finite incidence structure consisting of points and three pairwise disjoint classes of lines, each of size $n$, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the three classes. The current interest around $3$-nets (embedded) in a projective plane $PG(2,K)$, defined over a field $K$ of characteristic $p$, arose from algebraic geometry. It is not difficult to find $3$-nets in $PG(2,K)$ as far as $0 n$. Under this condition, the known families are characterized as the only $3$-nets in $PG(2,K)$ which can be coordinatized by a group. In this paper we deal with $3$-nets in $PG(2,K)$ which can be coordinatized by a diassociative loop $G$ but not by a group. We prove two structural theorems on $G$. As a corollary, if $G$ is commutative then every non-trivial element of $G$ has the same order, and $G$ has exponent $2$ or $3$. We also discuss the existence problem for such $3$-nets.

math.GR↗
arXiv2015

Large $p$-groups of automorphisms of algebraic curves in characteristic $p$

Let $S$ be a $p$-subgroup of the $\mathbb K$-automorphism group $Aut(\mathcal X)$ of an algebraic curve $\mathcal X$ of genus $g\ge 2$ and $p$-rank $γ$ defined over an algebraically closed field $\mathbb{K}$ of characteristic $p\geq 3$. Nakajima proved that if $γ\ge 2$ then $|S|\leq \textstyle\frac{p}{p-2}(g-1)$. If equality holds, $\mathcal X$ is a Nakajima extremal curve. We prove that if $$|S|>\textstyle\frac{p^2}{p^2-p-1}(g-1)$$ then one of the following cases occurs: (i) $γ=0$ and the extension $\mathbb K(\mathcal X)|\mathbb K(\mathcal X)^S$ completely ramifies at a unique place, and does not ramify elsewhere. (ii) $|S|=p$, and $\mathcal X$ is an ordinary curve of genus $g=p-1$. (iii) $\mathcal X$ is an ordinary, Nakajima extremal curve, and $\mathbb K(\mathcal X)$ is an unramified Galois extension of a function field of a curve given in (ii). There are exactly $p-1$ such Galois extensions. Moreover, if some of them is an abelian extension then $S$ has maximal nilpotency class. The full $\mathbb{K}$-automorphism group of any Nakajima extremal curve is determined, and several infinite families of Nakajima extremal curves are constructed by using their pro-$p$ fundamental groups.

math.AG↗
arXiv2015

A characterization of the Artin-Mumford curve

Let $\mathcal{M}$ be the Artin-Mumford curve over the finite prime field $\mathbb{F}_p$ with $p>2$. By a result of Valentini and Madan, $\mbox{Aut}_{\mathbb{F}_p}(\mathcal{M})\cong H$ with $H=(C_p\times C_p)\rtimes D_{p-1}$. We prove that if $\mathcal{X}$ is an algebraic curve of genus $g=(p-1)^2$ such that $\mbox{Aut}_{\mathbb{F}_p}(\mathcal{X})$ contains a subgroup isomorphic to $H$ then $\mathcal{X}$ is birationally equivalent over $\mathbb{F}_p$ to the Artin-Mumford curve $\mathcal{M}$.

math.AG↗
arXiv2012

Hermitian codes from higher degree places

Matthews and Michel investigated the minimum distances in certain algebraic-geometry codes arising from a higher degree place $P$. In terms of the Weierstrass gap sequence at $P$, they proved a bound that gives an improvement on the designed minimum distance. In this paper, we consider those of such codes which are constructed from the Hermitian function field. We determine the Weierstrass gap sequence $G(P)$ where $P$ is a degree 3 place, and compute the Matthews and Michel bound with the corresponding improvement. We show more improvements using a different approach based on geometry. We also compare our results with the true values of the minimum distances of Hermitian 1-point codes, as well as with estimates due Xing and Chen.

math.AG↗
arXiv2009

On the structure of 3-nets embedded in a projective plane

We investigate finite 3-nets embedded in a projective plane over a (finite or infinite) field of any characteristic p. Such an embedding is regular when each of the three classes of the 3-net comprises concurrent lines, and irregular otherwise. It is completely irregular when no class of the 3-net consists of concurrent lines. We are interested in embeddings of 3-nets which are irregular but the lines of one class are concurrent. For an irregular embedding of a 3-net of order n greater than 4 we prove that, if all lines from two classes are tangent to the same irreducible conic, then all lines from the third class are concurrent. We also prove the converse provided that the order n of the 3-net is smaller than p. In the complex plane, apart from a sporadic example of order n=5 due to Stipins, each known irregularly embedded 3-net has the property that all its lines are tangent to a plane cubic curve. Actually, the procedure of constructing irregular 3-nets with this property works over any field. In positive characteristic, we present some more examples for n greater than 4 and give a complete classification for n=4.

math.CO↗