On the threefold minimal model program in positive and mixed characteristic
File(s)
Author(s)
Stigant, Liam
Type
Thesis
Abstract
This dissertation explores the Minimal Model Program (MMP) in positive and mixed
characteristic in dimension three with a particular focus on outputs of the program. In
purely positive characteristic we combine the program with a detailed study of conic
bundles to prove a birational boundedness result. We show that given a suitable set of
log Calabi-Yau varieties, we can construct a bounded family containing bres birational
to any member of the chosen set.
For threefolds over a base of dimension at least one, we resolve the Abundance Conjecture
for klt pairs in joint work with F. Bernasconi and I. Brivio. Showing in particular that
every klt minimal model in mixed characteristic admits an Iitaka Fibration. This is then
applied to prove an Invariance of Plurigenera result for suitable families of surfaces.
Finally we consider outstanding questions around Mori brations in mixed characteristic.
We show that every klt threefold MMP terminates and that any two Mori bre space
outputs of an MMP from the same starting pair are connected by a series of Sarkisov
links. As part of this we prove a mixed characteristic Finiteness of Minimal Models result.
While the proof is focused in dimension three, the arguments work in any generality given
that the requisite MMP results are known.
characteristic in dimension three with a particular focus on outputs of the program. In
purely positive characteristic we combine the program with a detailed study of conic
bundles to prove a birational boundedness result. We show that given a suitable set of
log Calabi-Yau varieties, we can construct a bounded family containing bres birational
to any member of the chosen set.
For threefolds over a base of dimension at least one, we resolve the Abundance Conjecture
for klt pairs in joint work with F. Bernasconi and I. Brivio. Showing in particular that
every klt minimal model in mixed characteristic admits an Iitaka Fibration. This is then
applied to prove an Invariance of Plurigenera result for suitable families of surfaces.
Finally we consider outstanding questions around Mori brations in mixed characteristic.
We show that every klt threefold MMP terminates and that any two Mori bre space
outputs of an MMP from the same starting pair are connected by a series of Sarkisov
links. As part of this we prove a mixed characteristic Finiteness of Minimal Models result.
While the proof is focused in dimension three, the arguments work in any generality given
that the requisite MMP results are known.
Version
Open Access
Date Issued
2022-07
Date Awarded
2022-10
Copyright Statement
Creative Commons Attribution NonCommercial Licence
License URL
Advisor
Cascini, Paolo
Sponsor
Engineering and Physical Sciences Research Council (EPSRC)
Grant Number
EP/L015234/1
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)