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Hilbert polynomials of moduli spaces of rank 2 vector bundles. I. (English) Zbl 0798.14003

Let \(C\) be a smooth complex curve of genus \(g>1\), \(\xi\) be a line bundle of even degree on \(C\). The moduli space \({\mathcal M}^ C_ 0\) is the space of isomorphism classes of rank 2 semi-stable vector bundles over \(C\), such that \(\text{det} V = \xi\). The space \({\mathcal M}_ 1^ C\) is defined similarly with \(\xi\) replaced by an odd degree line bundle. These spaces can be given the structure of projective algebraic varieties of dimension \(3g-3\). \({\mathcal M}^ C_ 1\) is smooth while \({\mathcal M}^ C_ 0\) is singular, except for \(g = 2\). The Picard groups of \({\mathcal M}^ C_ 0\) and \({\mathcal M}^ C_ 1\) are freely generated over \(\mathbb{Z}\) by one ample generator. Let this ample line bundle be \(L_ 0\) for \({\mathcal M}_ 0\) and \(L_ 1\) for \({\mathcal M}_ 1\). In the present paper the author proves the following formula \(\dim H^ 0({\mathcal M}^ C_ 1, L^ k_ 1) = \sum^{2k + 1}_{j=1} (-1)^{j+1} ((k + 1)/ \sin^ 2(j \pi/2k + 2))^{g-1}\).
[See also part II of this paper: A. Bertram and the author, ibid. 32, No. 3, 599-609 (1993; see the following review)].

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series

Citations:

Zbl 0798.14004
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