Mukai’s program for curves on a K3 surface (original) (raw)

Let C be a general element in the locus of curves in Mg lying on some K3 surface, where g is congruent to 3 mod 4 and greater than or equal to 15. Following Mukai's ideas, we show how to reconstruct the K3 surface as a Fourier-Mukai transform of a Brill-Noether locus of rank two vector bundles on C. Contents 1. Introduction 1 2. Moduli of sheaves on a K3 surface 4 3. Brill-Noether loci for moduli of vector bundles on C 6 4. Geometry of (S, C) in the Rank-2 case 9 5. Brill-Noether loci in the Rank-2 case 12 6. Brill-Noether loci in the Rank-1 case 21 7. The Fourier-Mukai transform 22 References 26 In our work we take Mukai's paper [10] as a blueprint and generalize it to all genera which are congruent to 3, mod 4 and greater or equal than 11. Let (S, C) be a general point in P g , with g = 2s + 1, s ≥ 5 odd. Mukai's strategy to reconstruct the surface S from the curve C is as follows: consider the Brill-Noether locus M C (2, K C , s) which is the moduli space of semistable rank-two vector bundles on C having canonical determinant and possessing at least s + 2 linearly independent sections. Then M C (2, K C , s) is a K3 surface and the surface S can be obtained as an appropriate Fourier-Mukai transform of it. When g = 11 the proof consists of three main steps. One first considers pairs (S ′ , C ′) where S ′ is a K3 surface of a special type, and proves with ad-hoc constructions that M C ′ (2, K C ′ , s) is isomorphic to S ′ by showing that both are isomorphic to a moduli space M v (S ′) of vector bundles on S ′. The second step consists in deforming (S ′ , C ′) to a general pair (S, C): since M C (2, K C , 5) has expected dimension equal to two it is a flat deformation of M C ′ (2, K C ′ , s), thus it is again a K3 surface. Finally one shows the existence of an appropriate polarization h on M C (2, K C , 5) which induces an isomorphism between S and the Fourier-Mukai transform of M C (2, K C , 5) with respect to h. The first difficulty in trying to extend this proof is that when g = 2s + 1, s ≥ 6, the expected dimension of M C (2, K C , s) is zero for s = 6 and negative for s ≥ 7, so that it is not even clear that M C (2, K C , s) is non-empty when s ≥ 7. However, in her paper [15], Voisin associates a rank-two vector bundle E L to each base-point-free pencil |L| on C of degree s + 2. Each of these bundles is exhibited as an extension 0 → K C L −1 → E L → L → 0 and one can prove that Voisin's bundles E L are stable, (see, for instance, Lemma 2.5, Proposition 3.1 and Remark 5.11) and that, as L varies in W 1 s+2 (C), they describe a one-dimensional locus in M C (2, K C , s).