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Kodaira dimension of moduli space of vector bundles on surfaces. (English) Zbl 0799.14015

Let \(H\) be an ample line bundle on a smooth algebraic complex surface \(X\) and let \(M_ H\) be the moduli space of rank-2 \(H\)-semistable sheaves \(E\) on \(X\) with fixed determinant and given second Chern class. The dimension of \(M_ H\), its singularities, its normality, and its Kodaira dimension are studied in this paper, in line with the work done in understanding the geometry of \(M_ H\) by Gieseker, Maruyama, Donaldson, Friedman and others. As to the Kodaira dimension of \(M_ H\), known results strongly suggest that it should always be closely connected with the Kodaira dimension of \(X\). More precisely these dimensions are the same if \(X = \mathbb{P}^ 2\) (Barth, Hulek and others), for some ruled surfaces (Quin) as well as for K3 surfaces (Mukai), whereas \(\kappa (M_ H) \geq 0\) for some surfaces of general type (O’Grady). In this paper the author shows \(\kappa (M_ H) = 2\) for a class of minimal surfaces of general type.

MSC:

14J10 Families, moduli, classification: algebraic theory
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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References:

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