Title: Uniruledness of M11, and prime Fano 3-folds V22 of genus 12

Abstract: Let Mg be the moduli space of curves of genus g >1 and Kg the locus of those which are embeddable in K3 surfaces. Kg is of expected dimension min{3g-3, g+19} except for g = 10, 12. In the case of genus 11, this implies the uniruledness of M11 (Mori-M. 1983). In the case of genus 12, 30-dimensional K12 contains four subloci [Clebsch-Luroth], [2.7], [3.5] and [4.4], corresponding to four 1-nodal degenerations of V22. If a curve C in K12 belong to none of these loci, then C is the intersection of two anticanonical members of a Fano 3-fold V22 in a unique way (the last linear section theorem).