Determinant line bundles on Moduli spaces of pure sheaves on rational surfaces and Strange Duality

2010/05/18 by Yao Yuan, Yuan, Yao
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.1005.3201

arxiv created 2010/07/24 · arxiv updated 2010/07/27

Abstract

Let \mhu be the moduli space of semi-stable pure sheaves of class u on a smooth complex projective surface X. We specify u=(0,L,χ(u)=0), i.e. sheaves in u are of dimension 1. There is a natural morphism π from the moduli space \mhu to the linear system \ls. We study a series of determinant line bundles \lcn on \mhu via π. Denote g_L the arithmetic genus of curves in \ls. For any X and g_L≤0, we compute the generating function Z^r(t)=∑_nh⁰(\mhu,\lcn)tⁿ. For X being ℙ² or ℙ(\mo_\pone⊕\mo_\pone(-e)) with e=0,1, we compute Z¹(t) for g_L>0 and Z^r(t) for all r and g_L=1,2. Our results provide a numerical check to Strange Duality in these specified situations, together with Göttsche's computation. And in addition, we get an interesting corollary in the theory of compactified Jacobian of integral curves.

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