Member Seminar

Speaker: Iacopo Brivio (CMSA)

Title: Abundance for mixed characteristic threefolds

Abstract: The Minimal Model Program (MMP) predicts that every algebraic variety X is birational to either a fibration in Fano varieties, or it admits a “minimal model” X’, that is a birational model with nef canonical bundle K_X’. The Abundance conjecture predicts then that K_X’ is actually semiample, in particular it endows X’ with the structure of a Calabi-Yau fibration. These conjectures were initially phrased for complex varieties, but more recently there has been a lot of interest in working over positive characteristic fields, or even mixed characteristic rings. In this talk I will give a broad overview of the subject, starting from the case of complex surfaces. In the last part I will outline a proof of the Abundance conjecture for mixed characteristic threefolds (based on joint work with F. Bernasconi and L. Stigant).