Chamber Structure of Polarizations and the Moduli of Stable Sheaves on a Ruled Surface

LetX be a smooth projective surface defined over C andH an ample divisor onX. LetMH(r; c1, c2) be the moduli space of stable sheaves of rank r whose Chern classes (c1, c2) ∈ H (X,Q)×H(X,Q) and MH(r; c1, c2) the Gieseker-Maruyama compactification of MH(r; c1, c2). When r = 2, these spaces are extensively studied by many authors. When r ≥ 3, Drezet and Le-Potier [D1],[D-L] investigated the structure of moduli spaces on P, and Rudakov [R] treated moduli spaces on P×P. In this paper, we shall consider moduli spaces of rank r ≥ 3 on a ruled surface which is not rational. In particular, we shall compute the Picard group of MH(r; c1, c2). Let π : X → C be the fibration, f a fibre of π and C0 a minimal section of π with (C 2 0) = −e. We assume that e > 2g− 2, where g is the genus of C. Then KX is effective, and hence (KX , H) < 0 for any ample divisor H . In particular, MH(r; c1, c2) is smooth with the expected dimension 2r ∆− r(1− g) + 1. In section 2, we shall generalize the chamber structure of Qin [Q2]. As an application, we shall consider the difference of Betti numbers of moduli spaces on a ruled surface. Although we cannot generalize the method in [Y2, 0] directly, by using Qin’s method we can generalize it to any rank case. In [Y2], we computed the number of μ-semi-stable sheaves of rank 2 on a ruled surface defind over Fq. So, in principle, we can compute the Betti numbers of MH(3; c1, c2) on P . Combining chamber structure with another method, Göttsche [Gö] also considered the difference of Hodge numbers (and hence Betti numbers) of moduli spaces of rank 2. Matsuki and Wentworth [M-W] also generalized the chamber structure of polarizations. Combining another chamber structure, they showed that the rational map between two moduli spaces is factorized to a sequence of flips. In sections 4 and 5, we assume that X is a ruled surface which is not rational. Then, in the same way as in [Q1], we can give a condition for the existence of stable sheaves. Since we had computed the Picard group Pic(MH(r; c1, c2)) in case of (c1, f) = 0 [Y3], we assume that 0 < (c1, f) < r. In section 5, we shall compute the Picard group of MH(r; c1, c2), which is a generalization of [Q1] to r ≥ 3. The proof is the same as that in [D-N]. As is well known, it is difficult to treat the moduli spaces on rational ruled surfaces (cf. [D-L], [R]). However we can also check that MH(r; c1, c2) is emply or not in principle. I would like to thank Professor S. Mori for valuable suggestions.