Showing papers by "Gábor Korchmáros published in 2016"
TL;DR: In this article, the Nakajima bound was improved to a constant constant of 34 for the case of odd characteristic of the automorphisms of a projective, geometrically irreducible, nonsingular algebraic curve.
Abstract: 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 $c$.
14 citations
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TL;DR: In this article, it was shown that the Artin-Schreier-Mumford (ASM) curve has a semidirect automorphism group of order 2q^2(q-1)
Abstract: For a power $q$ of a prime $p$, the Artin-Schreier-Mumford curve $ASM(q)$ of genus $g=(q-1)^2$ is the nonsingular model $\mathcal{X}$ of the irreducible plane curve with affine equation $(X^q+X)(Y^q+Y)=c,\, c
eq 0,$ defined over a field $\mathbb{K}$ of characteristic $p$. The Artin-Schreier-Mumford curves are known from the study of algebraic curves defined over a non-Archimedean valuated field since for $|c|<1$ they are curves with a large solvable automorphism group of order $2(q-1)q^2 =2\sqrt{g}(\sqrt{g}+1)^2$, far away from the Hurwitz bound $84(g-1)$ valid in zero characteristic. In this paper we deal with the case where $\mathbb{K}$ is an algebraically closed field of characteristic $p$. We prove that the group $Aut(\mathcal{X})$ of all automorphisms of $\mathcal{X}$ fixing $\mathbb{K}$ elementwise has order $2q^2(q-1)$ and it is the semidirect product $Q\rtimes D_{q-1}$ where $Q$ is an elementary abelian group of order $q^2$ and $D_{q-1}$ is a dihedral group of order $2(q-1)$. For the special case $q=p$, this result was proven by Valentini and Madan. Furthermore, we show that $ASM(q)$ has a nonsingular model $\mathcal{Y}$ in the three-dimensional projective space $PG(3,\mathbb{K})$ which is neither classical nor Frobenius classical over the finite field $\mathbb{F}_{q^2}$.
2 citations
TL;DR: In this paper, it was shown that the only 3-net families in a projective plane can be coordinatized by a diassociative loop but not by a group, and if the loop is commutative then every non-trivial element of the loop has the same order, and every element has exponent either 2 or 3.
Abstract: 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.
TL;DR: This paper deals with 3-nets in PG(2,\mathbb {K}) which can be coordinatized by a diassociative loop G but not by a group, and proves two structural theorems on G.
Abstract: A 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 $$\mathrm{PG}(2,\mathbb {K})$$PG(2,K), defined over a field $$\mathbb {K}$$K of characteristic p, arose from algebraic geometry; see Falk and Yuzvinsky (Compos Math 143:1069---1088, 2007), Miguel and Buzunariz (Graphs Comb 25:469---488, 2009), Pereira and Yuzvinsky (Adv Math 219:672---688, 2008), Yuzvinsky (140:1614---1624, 2004), and Yuzvinsky (137:1641---1648, 2009). It is not difficult to find 3-nets in $$\mathrm{PG}(2,\mathbb {K})$$PG(2,K) as far as $$0n$$p>n. Under this condition, the known families are characterized as the only 3-nets in $$\mathrm{PG}(2,\mathbb {K})$$PG(2,K) which can be coordinatized by a group; see Korchmaros et al. (J Algebr Comb 39:939---966, 2014). In this paper we deal with 3-nets in $$PG(2,\mathbb {K})$$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 where the exponent of a finite diassociative loop is the maximum of the orders of its elements. We also discuss the existence problem for such 3-nets.