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.     

Construction of hermitian modular forms of genus 2 from cusp forms of genus 1

One constructs an integral operator, mapping the cusp modular forms of one variable into modular forms relative to Hermitian groups of genus 2 over an imaginary quadratic field. One computes explicitly the Fourier coefficients of the obtained forms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 51–67, 1985.     

Extreme rays for the large cone of hermitian generalized matrix functions

Given a <artwork name="GLMA31007ei1">-valued function f with domain <artwork name="GLMA31007ei2">, the symmetric group on {1,2,…, m}, we define the generalized matrix function [ f ](?), or d_f (?), in the usual way on the set of all m× m complex matrices. Letting <artwork name="GLMA31007ei3"> denote the set of all m× m positive semi-definite Hermitian matrices we consider the cone K _m whose elements are the Hermitian functions <artwork name="GLMA31007ei4"> such that [ f ]( A)≥0 for all <artwork name="GLMA31007ei5">. The extreme rays in K _m are fundamental to an understanding of the linear inequalities that result by restricting the generalized matrix functions [ f ](?) to the sets <artwork name="GLMA31007ei6">. In particular, the resolution of Lieb's permanent dominance conjecture, and certain similar conjectures such as the conjecture of Soules, will likely require identification and careful analysis of these rays. Grone, Merris, and Watkins have shown that the determinant function det(?), which is [ f ](?) if f is the signum function, is extreme in K _m for each m. We identify additional rays that are extreme for all m. In particular, we associate with each 2-term partition <artwork name="GLMA31007ei7"> of {1,2,…, m} an element <artwork name="GLMA31007ei8"> that is shown to be extreme in K _m for each m. If <artwork name="GLMA31007ei9"> is trivial, then <artwork name="GLMA31007ei10"> reduces to the determinant function; hence, our results are a natural extension of the result of Grone, Merris, and Watkins. Moreover <artwork name="GLMA31007ei11">, like det ( A), is expressible as a function of the eigenvalues of certain matrices related to A. Additional classes of extreme rays are also presented.     

On the genus of a maximal curve

The upper limit and the first gap in the spectrum of genera of -maximal curves are known, see [34], [16], [35]. In this paper we determine the second gap. Both the first and second gaps are approximately constant times , but this does not hold true for the third gap which is just 1 for while (at most) constant times q for This suggests that the problem of determining the third gap which is the object of current work on -maximal curves could be intricate. Here, we investigate a relevant related problem namely that of characterising those -maximal curves whose genus is equal to the third (or possible the forth) largest value in the spectrum. Our results also provide some new evidence on -maximal curves in connection with Castelnuovo's genus bound, Halphen's theorem, and extremal curves. Received: 1 January 2001 / Revised version: 30 July 2001 / Published online: 23 May 2002     

Curves of medium genus with many points

The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil–Serre upper bound for the number of points on the curve. We present algorithms for constructing curves of genus 5, 6, and 7 with small defect. Our aim is to be able to produce, in a reasonable amount of time, curves that can be used to populate the online table of curves with many points found at manypoints.org.     

Hyperbolicity of the genus two separating curve complex

In this short note, using a result of Brock-Masur, we show that the genus two separating curve complex is hyperbolic in the sense of Gromov, answering a conjecture of S. Schleimer in the affirmative.     

Curves of maximum genus in range A and stick-figures

In this paper we show the existence of smooth connected space curves not contained in a surface of degree less than , with genus maximal with respect to the degree, in half of the so-called range A. The main tool is a technique of deformation of stick-figures due to G. Fløystad.

     

Graphs of given genus and arbitrarily large maximum genus

The maximum genus of a connected graph G is the maximum among the genera of all compact orientable 2-manifolds upon which G has 2-cell embeddings. In the theorems that follow the use of an edge-adding technique is combined with the well-known Edmonds' technique to produce the desired results. Planar graphs of arbitrarily large maximum genus are displayed in Theorem 1. Theorem 2 shows that the possibility for arbitrarily large difference between genus and maximum genus is not limited to planar graphs. In particular, we show that the wheel graph, the standard maximal planar graph, and the prism graph are upper embeddable. We then show that given m and n, there is a graph of genus n and maximum genus larger than mn.     

On the genus of a cyclic curve over a finite field

Cyclic curves, i.e. curves fixed by a cyclic collineation group, play a central role in the investigation of cyclic arcs in Desarguesian projective planes. In this paper, the genus of a cyclic curve arising from a cyclic k-arc of Singer type is computed.     

Remarks on ample vector bundles of curve genus two

Let be an ample vector bundle of rank n – 1 on a smooth complex projective variety X of dimension n≥ 3 such that X is a -bundle over and that for any fiber F of the bundle projection . The pairs with = 2 are classified, where is the curve genus of . This allows us to improve some previous results. Received: 13 June 2006     

On the u-invariant of hermitian forms

Let K be a field of characteristic not 2 and A a central simple algebra with an involution σ. A result of Mahmoudi provides an upper bound for the u-invariants of hermitian forms and skew-hermitian forms over (A, σ) in terms of the u-invariant of K. In this paper we give a different upper bound when A is a tensor product of quaternion algebras and σ is a the tensor product of canonical involutions. We also show that our bounds are sharper than those of Mahmoudi.