International Congress of Mathematicians

Vector Bundles on a K3 Surface 499Assume that G lies on a K3 surface S. If F belongs to Ms(r, L, s), then therestriction F|c is of canonical determinant and we have h°(E\c) > h°(E) > x(E) =r + s. So E\c belongs to Slic(2,K : r + s) if it is stable. This is one motivationof the above definition. The case of genus 11, the gap value of genera where Fano3-folds of the next section do not exist, is the moist interesting.Theorem 3 ([15]) If C is a general curve of genus 11, then the Brill-Noether locusT = Slic(2,K : 7) of type III is a K3 surface and the restriction L of thedeterminant line bundle is of degree 20.There exists a universal family £ on G x T. We moreover have the following:• the restriction £\ XX T is is stable and belongs to Mr(2, L, 5), for every x £ C,and• the classification morphism G —• T = MT(2,L,5) is an embedding.This embedding is a non-Abelian analogue of the Albanese morphism X —•Pico(PicoX) and we have the following:Corollary A general curve of genus eleven has a unique embedding to a K3 surface.In [9], we studied the forgetful map ip g from the moduli space V g of pairs ofa curve G of genus g and a K3 surface S with C C S to the moduli space M g ofcurves of genus g and the generically finiteness of ipn. The above correspondenceG H> T gives the inverse rational map of ipn. We recall the fact that ipio is notdominant in spite of the inequality dimoio = 29 > dim .Mio = 27 ([12]).6. Fano 3-foldsA smooth 3-dimensional projective variety is called a Fano 3-fold if the anticanonicalclass — Kx is ample. In this section we assume that the Picard groupPicX is generated by —Kx- The self intersection number (—Kx) 3 = 2g — 2 isalways even and the integer g > 2 is called the genus, by which the Fano 3-folds areclassified into 10 deformation types. The values of g is equal to 2,..., 10 and 12. AFano 3-folds of genus g < 5 is a complete intersection of hypersurfaces in a suitableweighted projective space.By Shokurov [25], the anticanonical linear system | — Kx\ always contain asmooth member S, which is a K3 surface. In [13] we classified Fano 3-folds X ofPicard number one using rigid bundles, that is, F £ Ms(r,L,s) with (F 2 ) — 2rs =—2. For example X is isomorphic to a linear section of the 10-dimensional spinorvariety, that is,x~Si 3 nffin"-nff 7) (2)in the case of genus 7 and a linear sectionx ~Zi 6 nHinH 2 nH 3 , (3)of the 6-dimensional symplectic, or Lagrangian, Grassmann variety Sie = SP(6)/U(3) C P 13 in the case of genus 9. The non-Abelian Brill-Noether loci shed newlight on this classification.