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V. Mercat 《manuscripta mathematica》1999,98(1):75-85
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 相似文献
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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. 相似文献
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Discrete Riemann Surfaces and the Ising Model 总被引:1,自引:0,他引:1
Christian Mercat 《Communications in Mathematical Physics》2001,218(1):177-216
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 相似文献
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