Convex Fujita numbers

This is an account of an ongoing research project with Yusuf Mustopa and Jakob Stix. We study effective global generation properties of adjoint line bundles, nevertheless in a somewhat more extended setting than considered traditionally. Originally motivated by Fujita’s conjecture on global generation, we investigate when adjoint line bundles coming from sums of ample ones become globally generated. It turns out that there is a lot to say about the resulting new invariants dubbed convex Fujita numbers. I will discuss a variety of cases when the answer is known, and directions for future research.