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

This quantity includes 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 commemorated S. Ramanan on his seventieth birthday. the most components lined during this quantity are vector bundles, parabolic bundles, abelian forms, Hilbert schemes, touch constructions, index concept, Hodge conception, and geometric invariant thought. Professor Ramanan has made vital contributions in all of those parts

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**Vector Bundles and Complex Geometry**

This quantity features 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 constructions, index conception, Hodge thought, and geometric invariant idea.

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5]. Since U U L L −1 ∼ ˜ implies that Pic UL → Pic (p UL ) = Pic UL . Since UL is projective, so is ˜L ) ≥ 1, rank (Pic U ) ≥ 1. 4], we have Pic UL → Pic ULs . Thus rank (Pic ULs ) ≥ 1. 3 that Pic ULs ∼ = Z and hence Pic UL ∼ = Z. The variety UY (2, d) has a ﬁltration U := UY (2, d) = W2 ⊃ W1 ⊃ W0 40 10 USHA N. BHOSLE where Wi are seminormal, the non-normal locus of Wi is Wi−1 for i = 1, 2 and W0 ∼ = UX (2, d − 2) is smooth. 2]). 10]. 5. [B4, Theorem 2 and Theorem 3B] Let gY ≥ 2 and let L be a torsionfree sheaf of rank 1 on Y which is not locally free.

This means that the conic x2 − αy 2 − βz 2 = 0 is isomorphic to P1K , thus there exists λ, μ in K with λ2 − α − βμ2 = 0. We choose α and β in Ks such that α 2 = α and β 2 = β; as above we deﬁne the homomorphisms a and b : g → Z/2 by σ α = (−1)a(σ) α and σ β = (−1)b(σ) β for each σ ∈ g . 28 6 ARNAUD BEAUVILLE z−θ βμ λ−α ; let g ∈ PGL2 (Ks ) be the homography z → α . = λ+α βμ z+θ An easy computation gives Put θ := g −1 σg = i(a(σ), b(σ)) . Thus the embedding of (Z/2)2 associated to (α, β) is gig −1 ; it maps e1 to the λu − α , and e2 to h2 : z → α/z .

Hence RΓs is dense in RΓss . 13. If the second Chern class c2 of the underlying bundles is large, then Msα is dense in Mss α for any weight α. Proof. By the Seshadri–Biswas correspondence we see that the moduli stack s ss Mss Γ of Γ–semistable bundles (resp. MΓ of Γ–stable bundles) is isomorphic to Mα s s (resp. Mα ). 12, since c2 0 it follows that RΓ is dense in s RΓss . Since RΓss (resp. RΓs ) is the atlas of the Artin stack Mss Γ (resp. MΓ ) we have ss ss a surjective morphism RΓ → MΓ and the result follows.

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