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Picard groups of the moduli spaces of vector bundles, Math. Ann. 314 (1999), 245–263. [B4] Bhosle Usha N. Picard groups of the moduli spaces of semistable sheaves, Proc. Indian Acad. Sci. (Math. ) 114 (2004), no. 2, 107–122. [B5] Bhosle Usha N. Seminormal varieties, torsionfree sheaves, and Picard groups, Communications in Algebra, 36 (2008), no. 3, 821–841. S. Groupe de Picard des varietes de modules de ﬁbres semistables sur les courbes algebriques, Invent. Math. 97 (1989), 53–94. , Gagn´ e M.

Variation of Mss α and the Main Theorem s Let Mss α (resp. Mα ) be the moduli stack of α-semistable (resp. α-stable) bundles with ﬁrst Chern class 0. 7). Now we study these moduli stacks as we vary weights. Let N0 = D · H, W = {(α1 , α2 ) : 0 < α1 < α2 < 1} and δW := (α1 , α2 ) : 0 < α1 < α2 < 1 such that | α1 − α2 |= k , 1 ≤ k ≤ 2N0 . 2N0 Let W ◦ = W − δW A connected component of W ◦ is called a chamber. Observe s that, if α is a weight within a chamber then Mss α = Mα . Moreover if α and β are ss ss s s in same chamber then Mα = Mβ and also Mα = Mβ .

Algebraic Geometry. Graduate Texts in Mathematics 52. Springer-Verlag, New York-Heidelberg, 1977. [9] D. Huybrechts, M. Lehn. The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedr. Vieweg und Sohn, Braunschweig, 1997. [10] Y. Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), 253–276. [11] Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem, Algebraic Geometry, Sendai, 1985, 283–360, Adv. Stud. Pure. Math. 10, North-Holland, Amsterdam, 1987.