Stability and moduli spaces of syzygy bundles
[eng] To determine whether a syzygy bundle on PN is stable, or semistable, is a long-standing problem in algebraic geometry. It is closely related to the problem of finding the Hilbert function and the minimal free resolution of the coordinate ring of the variety defined by a family of general homog...
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| Formato: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2009 |
| País: | España |
| Recursos: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/35146 |
| Acesso em linha: | https://hdl.handle.net/2445/35146 http://www.tdx.cat/TDX-0611110-111551 http://hdl.handle.net/10803/669 |
| Access Level: | acceso abierto |
| Palavra-chave: | Teoria de mòduls Geometria algebraica Moduli theory Algebraic geometry |
| Resumo: | [eng] To determine whether a syzygy bundle on PN is stable, or semistable, is a long-standing problem in algebraic geometry. It is closely related to the problem of finding the Hilbert function and the minimal free resolution of the coordinate ring of the variety defined by a family of general homogeneous polynomials f1, . . . , fn in K[X0, . . . ,XN]. This problem goes back at least to the eighties, when Fröberg addresses it in his paper, to find a lower estimate for the Hilbert series of such a ring in terms of the degrees of f1, . . . , fn. In this thesis we consider the case of syzygy bundles defined by general forms f1, . . . , fn of the same degree d, and prove their stability and unobstructedness for N ≥ 2, except for the case (N, d, n) = (2, 2, 5), where only semistability is guaranteed. To this end, we focus on the case of monomials and derive consequences for general forms from here. The main goal of this work is therefore to give a complete answer to the following problem: Does there exist for every d and every n ≤ (d+N / N) a family of n monomials in K [X0, . . . ,XN] of degree d such that their syzygy bundle is semistable? |
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