Groups generated by spherical twists on K3 surfaces and full exceptional collections on Fano threefolds

Abstract

Let Y be a smooth K3 surface of Picard rank 1. We prove that the subgroup G of Aut D^b(Y) generated by spherical twists with respect to all spherical objects is free. Moreover, we provide a precise recipe to find free generators of G and determine the cases when G is finitely generated, depending on the degree of Y. This description in particular yields a precise classification of spherical objects in Aut D^b(Y). We apply these results to verify the first three-dimensional case of a conjecture due to Bondal and Polishchuck, namely, we establish the transitivity of the braid group action on full exceptional collections for Fano threefolds of Picard rank 1.

Many corrections, changes in the exposition, Theorem 6.7 added