By Oscar Garcia-Prada, Peter E. Newstead, Luis Álvarez-Cónsul, Indranil Biswas, Steven B. Bradlow, Tomás L. Gómez, Luis Alvarez-consul

This quantity encompasses a selection of papers from the convention on Vector Bundles held at Miraflores de l. a. Sierra, Madrid, Spain on June 16-20, 2008, which venerated S. Ramanan on his seventieth birthday. the most components lined during this quantity are vector bundles, parabolic bundles, abelian kinds, Hilbert schemes, touch buildings, index idea, Hodge thought, and geometric invariant conception. Professor Ramanan has made vital contributions in all of those components

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Let gY = 1. (i) If (n, d) = 1, then Pic UY (n, d) ∼ = Pic Y . If Y is singular, then Pic UY (n, d) ∼ = Gm ⊕ Z and Pic UY (n, d), Pic UL (n, d) are trivial. (ii) If h = (n, d) > 1 then Pic UY (n, d) ∼ = Pic S h (Y ). ∼ For L ∈ JY , Pic UL (2, 0) = Z. If Y is singular, then Pic UL (2, 0) and Pic UY (2, 0) are trivial. Proof. By the explicit description, Pic UY (n, d) ∼ = Pic S h (Y ). If (n, d) = 1, then UL (n, d) is a point, so its Picard group is trivial. If Y is nodal, then Pic Y ∼ = Gm ⊕ Z = Gm ⊕ Pic P1 ∼ 38 8 USHA N.

11]. 2. 17]. 8. Computing the image HU of Pic U in Pic W1 explicitly seems diﬃcult as we do not have enough information about Pic P1 . One has HU = {B ∈ Pic W1 | p∗ (B) = (Det∗ L + m LY ) |P1 , L ∈ G, m ∈ Z}. Here LY = p∗ LY , LY being the determinant line bundle on Y . Det denotes the d determinant morphism P → JX which associates to a GPB (E, Q) the determinant of the underlying vector bundle E. Note that every GPS is S-equivalent to a GPB, so there is a well-deﬁned morphism Det [Su]. 42 12 USHA N.

D] J. Dieudonn´e: La g´ eom´ etrie des groupes classiques. Springer-Verlag, Berlin-G¨ ottingenHeidelberg, 1955. [I] I. Isaacs: Character theory of ﬁnite groups. AMS Chelsea Publishing, Providence, RI, 2006. -P. Serre: Corps locaux. Hermann, Paris, 1962. -P. Serre: Propri´ et´ es galoisiennes des points d’ordre ﬁni des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331. -P. Serre: Galois cohomology. Springer-Verlag, Berlin, 1997. -A. fr This page intentionally left blank Contemporary Mathematics Volume 522, 2010 Picard Groups of Moduli Spaces of Torsionfree Sheaves on Curves Usha N.