January 9, 2019
TITLE: Prime Fano 3-folds and BN-general K3s

ABSTRACT: Fano 3-folds with 2nd Betti number one are classified into 17 deformation types. The anti-canonical degree 2g-2 and the 3rd Betti number 2p are their basic numerical invariants. The sum g+p varies from 12 to 54, and the minimum 12 is attained in 3 cases. In this talk I will explain the linear section theorem in the case (g,p)=(10,2): a prime Fano 3-fold of g=10 is obtained from the (5-dimensional) G2-adjoint variety by taking hyperplane section p=2 times. The basic tool is a rigid, or spherical, vector bundle on a K3 surface S in the anti-canonical linear system. The key property of S used in the proof is the BN-genericity.