Determinant line bundles on moduli spaces of pure sheaves on rational surfaces and strange duality

Abstract

Abstract. Let M H X (u) 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 M H X (u) to the linear system |L|. We study a series of determinant line bundles λ c r n on M H X (u) via π. Denote g L the arithmetic genus of curves in |L|. For any X and g L ≤ 0, we compute the generating function Z r (t) = n h 0 (M H X (u), λ c r n )t n . For X being P 2 or P(O P 1 ⊕ O P 1 (−e)) with e = 0, 1, we compute Z 1 (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 (Corollary 4.2.13) in the theory of compactified Jacobian of integral curves.

Strange Duality on Rational Surfaces II: Higher-Rank Cases

“…SD 1,L is surjective by Corollary 4.3.2 in [28]. Combining (4.11) and (4.13), we have proved the proposition for L = −K X .…”

“…Proof. If L = −K X , then D Θ L = ∅ by Proposition 4.1.1 and Corollary 4.3.2 in [28]. On the other hand, we want to show that H 0 (λ r (L ⊗ K X )) = 0.…”

“…Mainly there are two formulations for surfaces, one of which is due to Le Potier (see [21], [8] or §2.4 in [15]) for simply connected surfaces, while the other is due to Marian-Oprea for K3 and Abelian surfaces (see [23] or [25]). Both formulations have been studied by many people and the conjecture has been proven true for a number of cases ( [1], [2], [6], [7], [8], [15], [24], [25], [26], [28], [30], [31]). In spite of that, on strange duality for surfaces what we have known is still little.…”

Strange duality on rational surfaces

“…By Corollary 3.19, the strange duality map SD c 1 2 ,u L in (3.4) is a map between two vector spaces of same dimension, while L is in case (1) of the theorem. The argument proving Corollary 4.3.2 in [19] applies and hence SD c 1 2 ,u L is an isomorphism. Statement (1) is proved.…”

Affine pavings for moduli spaces of pure sheaves on $$\mathbb {P}^2$$ P 2 with degree $$\le 5$$ ≤ 5

“…If G is pure, then G ∨∨ G and moreover G and G ∨ are determined by each other (see [13] Lemma A.0.13). Any sheaf F = coker( f ) in W d has its dual F ∨ determined by the following sequence (see the diagram (4.2) in [14]).…”