1,720,989 research outputs found
Ordinary algebraic curves with many automorphisms in positive characteristic
No full textLet X be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus g (X) >= 2 defined over an algebraically closed field K of odd characteristic p. Let Aut(X) be the group of all automorphisms of X which fix K elementwise. For any solvable subgroup G of Aut(X) we prove that vertical bar G vertical bar cp(2) for some positive constant c
Strongly irreducible collineation groups on an oval in a projeclive plane of odd order (Preprint)
No full text
Multilevel secret sharing schemes arising from the normal rational curve
No full textMultilevel secret sharing schemes are used in cryptography when a responsibility has to be delegated to qualified groups of participants rather than to a single one. In the present paper we propose a mathematical model for such a scheme based on projective geometry over a finite field. The main ingredients are normal rational curves and their osculating subspaces
Large odd prime power order automorphism groups of algebraic curves in any characteristic
No full textLet X be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus g ≥ 2 defined over an algebraically closed field K of odd characteristic p ≥ 0, and let Aut(X) be the group of all automorphisms of X which fix K element-wise. For any a subgroup G of Aut(X) whose order is a power of an odd prime d other than p, the bound proven by Zomorrodian for Riemann surfaces is |G| ≤ 9(g − 1) where the extremal case can only be obtained for d = 3 and g ≥ 10. We prove Zomorrodian’s result for any K. The essential part of our paper is devoted to extremal 3-Zomorrodian curves X. Two cases are distinguished according as the quotient curve X/Z for a central subgroup Z of Aut(X) of order 3 is either elliptic, or not. For elliptic type extremal 3-Zomorrodian curves X, we completely determine the two possibilities for the abstract structure of G using deeper results on finite 3-groups. We also show infinite families of extremal 3-Zomorrodian curves for both types, of elliptic or non-elliptic. Our paper does not adapt methods from the theory of Riemann surfaces, nevertheless it sheds a new light on the connection between Riemann surfaces and their automorphism groups
A Characterization of the sharply 3-transitive finite permutation groups
No full textGeneralizing a result by N. Percsy, we prove a sufficient conditionfor a sharply 3-transitive finite permutation set with identity to be a group; the proof makes use of M. J. Kallaher's theorem on finite Bol quasifield
The geometry of the Artin--Schreier-Mumford curves over an algebraically closed field
No full textFor a power q of a prime p, the Artin–Schreier–Mumford curve ASM(q) of genus g = (q − 1)2 is the nonsingular model X of the irreducible plane curve with affine equation (Xq + X)(Y q + Y ) = c, c ̸= 0, defined over a field 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)q2 = 2√g(√g+1)2, far away from the Hurwitz bound 84(g−1) valid in zero characteristic; see [2–4]. In this paper we deal with the case where K is an algebraically closed field of characteristic p. We prove that the group Aut(X ) of all automorphisms of X fixing K elementwise has order 2q2(q − 1) and it is the semidirect product Q ⋊ Dq−1 where Q is an elementary abelian group of order q2 and Dq−1 is a dihedral group of order 2(q − 1). For the special case q = p, this result was proven by Valentini and Madan [13]; see also [1]. Furthermore, we show that ASM(q) has a nonsingular model Y in the three-dimensional projective space P G(3, K) which is neither classical nor Frobenius classical over the finite field Fq2
Algebraic Curves and maximal arcs
Full text linkA lower bound on the minimum degree of the plane algebraic curves
containing every point in a large point--set \cK of the
Desarguesian plane \PG(2,q) is obtained. The case where \cK is
a maximal --arc is considered in greater depth
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