On the structure of the effective cone of divisors

The Cone and Contraction Theorems describing the Kleiman- Mori cone of curves for smooth varieties, due to Mori and later generalized to a larger class of varieties, are pivotal results in algebraic geometry, in the context of projective varieties, playing an important role in the Minimal Model Program.

In this talk we ask if the pseudo-effective cone of divisors associated to a smooth projective variety inherits a structure theorem that resembles the one of Mori’s Cone Theorem. In this lecture, we focus on the blown up of projective space at a collection of general points, X. To conjecture the existence of such a structure theorem we will first define the canonical curve class for X. We later present evidence for this question, including a result of Isabel Stenger and Zhixin Xie.

This talk is based on work in progress with Rick Miranda, as well as Chiara Brambilla, Elisa Postinghel and Luis Jose Santana Sanchez and past work with Nathan Priddis.