Abstract
Let X be any smooth simply connected projective surface. We consider some moduli space of pure sheaves of dimension one on X, i.e. \({M_X^H(u)}\) with u = (0, L, χ(u) = 0) and L an effective line bundle on X, together with a series of determinant line bundles associated to \({r[\mathcal{O}_X]-n[\mathcal{O}_{pt}]}\) in the Grothendieck group of X. Let g L denote the arithmetic genus of curves in the linear system |L|. For g L ≤ 2, we give a upper bound of the dimensions of sections of these line bundles by restricting them to a generic projective line in |L|. Our result gives, together with Göttsche’s computation, a first step of a check for the strange duality for some cases for X a rational surface.
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Yuan, Y : Determinant line bundles on moduli spaces of pure sheaves on rational surfaces and strange duality. arXiv, 1005.3201
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Yuan, Y. Estimates of sections of determinant line bundles on Moduli spaces of pure sheaves on algebraic surfaces. manuscripta math. 137, 57–79 (2012). https://doi.org/10.1007/s00229-011-0457-6
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DOI: https://doi.org/10.1007/s00229-011-0457-6