A Note on Moduli of Vector Bundles on Rational Surfaces

Let (X,H) be a pair of a smooth rational surface X and an ample divisor H on X . Assume that (KX , H) < 0. Let MH(r, c1, χ) be the moduli space of semi-stable sheaves E of rk(E) = r, c1(E) = c1 and χ(E) = χ. To consider relations between moduli spaces of different invariants is an interesting problem. If (c1, H) = 0 and χ ≤ 0, then Maruyama [Ma2], [Ma3] studied such relations and constructed a contraction map φ : MH(r, c1, χ) → MH(r − χ, c1, 0). Moreover he showed that the image is the Uhlenbeck compactification of the moduli space of μ-stable vector bundles. In particular, he gave an algebraic structure on Uhlenbeck compactification which was topologically constructed before. After Maruyama’s result, Li [Li] constructed the birational contraction for general cases, by using a canonical determinant line bundle, and gave an algebraic structure on Uhlenbeck compactification. Although Maruyama’s method works only for special cases, his construction is interesting of its own. Let us briefly recall his construction. Let E be a semi-stable sheaf of rk(E) = r, c1(E) = c1 and χ(E) = χ. Then H (X,E) = 0 for i = 0, 2. We consider a universal extension