0 Ju n 20 02 DEFORMATIONS OF THE PICARD BUNDLE

00 1 Deformations of the Picard Bundle

Let X be a nonsingular algebraic curve of genus g ≥ 3, and M ξ the moduli space of stable vector bundles of rank n ≥ 2 and degree d with fixed determinant ξ over X such that n and d coprime and d > n(2g − 2). We assume that if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3. Let W ξ (L) denote the vector bundle over M ξ defined as the direct image π * (U ξ ⊗ p * 1 L) where U ξ is a universal vector bu...

Deformations of the Generalised Picard Bundle

Let X be a nonsingular algebraic curve of genus g ≥ 3, and let Mξ denote the moduli space of stable vector bundles of rank n ≥ 2 and degree d with fixed determinant ξ over X such that n and d are coprime. We assume that if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3, and suppose further that n0, d0 are integers such that n0 ≥ 1 and nd0 + n0d > nn0(2g − 2). Let E be a semistable vector bundle over ...

ar X iv : 0 80 4 . 15 58 v 2 [ m at h . N T ] 1 7 Ju n 20 08 K 3 surfaces with Picard rank 20

We determine all complex K3 surfaces with Picard rank 20 over Q. Here the NéronSeveri group has rank 20 and is generated by divisors which are defined over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface. We also a...

ar X iv : h ep - p h / 02 06 17 1 v 2 2 4 Ju n 20 02 IPHE 2002 - 009 June 24 , 2002 B 0 – B 0 MIXING

m at h . A G ] 1 6 Ju n 20 02 ON A CLASSICAL CORRESPONDENCE BETWEEN K 3 SURFACES

Let X be a K3 surface which is intersection of three (a net P) of quadrics in P. The curve of degenerate quadrics has degree 6 and defines a double covering of P K3 surface Y ramified in this curve. This is the classical example of a correspondence between K3 surfaces which is related with moduli of vector bundles on K3 studied by Mukai. When general (for fixed Picard lattices) X and Y are isom...