Chris Brav

Ping-pong and exceptional vector bundles



Abstract

We present a strategy for proving that full exceptional collections of vector bundles on projective n-space can be constructed from a standard collection of line bundles, reducing the question of constructibility to the problem of freeness of a certain finitely generated linear group. We use the ping-pong lemma of Fricke-Klein to solve this problem in low dimensions, thus providing a new proof of constructibility of exceptional collections in some cases. We expect a similar ping-pong argument to give constructibility on projective n-space and on some other Fano varieties of Picard rank one. Constructibility results are useful for understanding spaces of stability conditions associated to such varieties. This is joint work in progress with Hugh Thomas.