Fujita vanishing, sufficiently ample line bundles, and cactus varieties

Abstract: For a fixed projective manifold 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? 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.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html