Le problème de Brill–Noether et le théorème de Teixidor

Let C be a smooth projective algebraic curve of genus g. The Brill–Noether problem is to determine at which conditions on n, d and k, there exists a stable vector bundle of rank n, degree d and with at least k independent global sections. Teixidor gave a rather general answer about the existence in her paper [Tei]. The result was available only for a generic curve, in a very imprecise sense, and the proof was really difficult. We show here, that the result is in fact true for all curves and a little more. Received: 5 September 1997 / Revised version: 15 June 1998     

Discrete Riemann Surfaces and the Ising Model

We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy–Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality. Received: 23 May 2000/ Accepted: 21 November 2000     

Moduli Spaces of Coherent Systems of Small Slope on Algebraic Curves

Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E, V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent systems for 0 < d ≤ 2n. We show that these spaces are irreducible whenever they are nonempty and obtain necessary and sufficient conditions for nonemptiness.