Wojciech J. Gajda

For a fixed projective variety X, we say that a property P(L) (where L is a line bundle on X) is satisfied by sufficiently ample line bundles if there exists a line bundle M on X such that P(L) hold for any L with L-M ample. I will discuss which properties of line bundles are satisfied by the sufficiently ample line bundles - for example, can you figure out before the talk, whether a sufficiently ample line bundle must be very ample? The grandfather of such properties and a basic ingredient used to study this concept is Fujita's vanishing theorem, which is an analogue of Serre's vanishing for sufficiently ample line bundles. At the end of the talk I will define cactus varieties (an analogue of secant varieties) and sketch a proof that cactus varieties to sufficiently ample embeddings of X are (set-theoretically) defined by minors of matrices with linear entries. The topic is closely related to conjectures of Eisenbud-Koh-Stillman (for curves) and Sidman-Smith (for any varieties). The new ingredients are based on a joint work in preparation with Weronika Buczyńska and Łucja Farnik.