Generalising a characterisation of Hermitian curves

This article proves a characterisation of the classical unital that is a generalisation of a characterisation proved in 1982 by Lefèvre-Percsy. It is shown that if is a Buekenhout-Metz unital with respect to a line in such that a line of not through meets in a Baer subline, then is classical. An immediate corollary is that if is a unital in PG such that is Buekenhout-Metz with respect to two distinct lines, then is classical. Received 5 August 1999; revised 15 February 2000.     

Curves covered by the Hermitian curve

A family of maximal curves is investigated that are all quotients of the Hermitian curve. These curves provide examples of curves with the same genus, the same automorphism group and the same Weierstrass semigroup at a generic point, but that are not isomorphic.     

On the intersection of Hermitian curves and of Hermitian surfaces

Kestenband [Unital intersections in finite projective planes, Geom. Dedicata 11(1) (1981) 107–117; Degenerate unital intersections in finite projective planes, Geom. Dedicata 13(1) (1982) 101–106] determines the structure of the intersection of two Hermitian curves of PG(2,q²), degenerate or not. In this paper we give a new proof of Kestenband's results. Giuzzi [Hermitian varieties over finite field, Ph.D. Thesis, University of Sussex, 2001] determines the structure of the intersection of two non-degenerate Hermitian surfaces and of PG(3,q²) when the Hermitian pencil defined by and contains at least one degenerate Hermitian surface. We give a new proof of Giuzzi's results and we obtain some new results in the open case when all the Hermitian surfaces of the Hermitian pencil are non-degenerate.     

On Unitals ith Many Baer Sublines

We identify the points of PG(2, q) ith the directions of lines in GF(q ³), viewed as a 3-dimensional affine space over GF(q). Within this frameork we associate to a unital in PG(2, q) a certain polynomial in to variables, and show that the combinatorial properties of the unital force certain restrictions on the coefficients of this polynomial. In particular, if q = p ² where p is prime then e show that a unital is classical if and only if at least (q - 2) secant lines meet it in the points of a Baer subline.     

Complete arcs arising from a generalization of the Hermitian curve (extended abstract)

We investigate arcs, in projective planes over finite fields, arising from the set of rational points of a generalization of the Hermitian curve. The degree of the arcs is closely related to the number of rational points of a class of Artin-Schreier curves.     

A few uncaught universal Hermitian forms

We will complete the list of universal binary Hermitian forms over imaginary quadratic fields by investigating three Hermitian forms missed by previous researchers.

     

LDPC codes from the Hermitian curve

In this paper, we study the code which has as parity check matrix the incidence matrix of the design of the Hermitian curve and its (q + 1)-secants. This code is known to have good performance with an iterative decoding algorithm, as shown by Johnson and Weller in (Proceedings at the ICEE Globe com conference, Sanfrancisco, CA, 2003). We shall prove that has a double cyclic structure and that by shortening in a suitable way it is possible to obtain new codes which have higher code-rate. We shall also present a simple way to constructing the matrix via a geometric approach.      

A Hermitian Curvature Flow

Chinese Annals of Mathematics, Series B - A Hermitian curvature flow on a compact Calabi-Yau manifold is proposed and a regularity result is obtained. The solution of the flow, if exists, is a...     

A Combinatorial Algorithm Related to the Geometry of the Moduli Space of Pointed Curves

As pointed out in Arbarello and Cornalba (J. Alg. Geom. 5 (1996), 705–749), a theorem due to Di Francesco, Itzykson, and Zuber (see Di Francesco, Itzykson, and Zuber, Commun. Math. Phys. 151 (1993), 193–219) should yield new relations among cohomology classes of the moduli space of pointed curves. The coefficients appearing in these new relations can be determined by the algorithm we introduce in this paper.     

The Complete Determination of the Minimum Distance of Two-Point Codes on a Hermitian Curve

This is a continuation of the previous papers [3, 4, 5]. We finish determining the minimum distance of two-point codes on a Hermitian curve. Masaaki Homma: Partially supported by Grant-in-Aid for Scientific Research (15500017), JSPS. Seon Jeong Kim: Partially supported by Korea Research Foundation Grant (KRF-2004-041-C00016)     

Embedding of a Maximal Curve in a Hermitian Variety

Let X be a projective, geometrically irreducible, non-singular, algebraic curve defined over a finite field F q ² of order q ². If the number of F q ²-rational points of X satisfies the Hasse–Weil upper bound, then X is said to be F q ²-maximal. For a point P ₀ X(F q ²), let be the morphism arising from the linear series D: = |(q + 1)P ₀|, and let N: = dim(D). It is known that N 2 and that is independent of P ₀ whenever X is F q ²-maximal.     

Complete p-Descent for Jacobians of Hermitian Curves

Let X be the Fermat curve of degree q+1 over the field k of q² elements, where q is some prime power. Considering the Jacobian J of X as a constant abelian variety over the function field k(X), we calculate the multiplicities, in subfactors of the Shafarevich–Tate group, of representations associated with the action on X of a finite unitary group. J is isogenous to a power of a supersingular elliptic curve E, the structure of whose Shafarevich–Tate group is also described.     

Derivation techniques on the Hermitian surface

We discuss derivation‐like techniques for transforming one locally Hermitian partial ovoid of the Hermitian surface H(3,q²) into another one. These techniques correspond to replacing a regulus by its opposite in some naturally associated projective 3‐space PG(3,q) over a square root subfield. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 478–486, 2007     

Projective Representations and Pure Pseudorepresentations of Hermitian Symmetric Simple Lie Groups

It is shown that an arbitrary irreducible continuous unitary projective representation of a simple Hermitian symmetric Lie group is generated by a strongly continuous pure unitary pseudorepresentation of the adjoint group of the Lie group.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 140–146.Original Russian Text Copyright © 2005 by A. I. Shtern.     

Pure Weierstrass gaps from a quotient of the Hermitian curve

In this paper, by employing some results on Kummer extensions, we give an arithmetic characterization of pure Weierstrass gaps at many totally ramified places on a quotient of the Hermitian curve, including the well-studied Hermitian curve as a special case. The cardinality of these pure gaps is explicitly investigated. In particular, the numbers of gaps and pure gaps at a pair of distinct places are determined precisely, which can be regarded as an extension of the previous work by Matthews (2001) considered Hermitian curves. Additionally, some concrete examples are provided to illustrate our results.