• Corpus ID: 229924399

Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic

  title={Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic},
  author={Bhargav Bhatt and Linquan Ma and Zsolt Patakfalvi and Karl Schwede and Kevin Tucker and Joe Waldron and Jakub Witaszek},
We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global F -regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita’s conjecture to mixed characteristic. 
Resolution and alteration with ample exceptional divisor
In this short note we explain how to construct resolutions or regular alterations admitting an ample exceptional divisor, assuming the existence of projective resolutions or regular alterations. In
Relative mmp without $ \mathbb{Q} $-factoriality
The main applications are to log terminal singularities, removing the earlier Q -factoriality assumption from several theorems of Hacon-Witaszek and de Fernex-Kollar-Xu.
An analogue of adjoint ideals and PLT singularities in mixed characteristic
We use the framework of perfectoid big Cohen-Macaulay (BCM) algebras to define a class of singularities for pairs in mixed characteristic, which we call purely BCM-regular singularities, and a
On the non-archimedean Monge-Amp\`ere equation in mixed characteristic
Let X be a smooth projective variety over a complete discretely valued field of mixed characteristic. We solve non-archimedean Monge– Ampère equations on X assuming resolution and embedded resolution
Algebraic geometry in mixed characteristic
Fix a prime number p. We report on some recent developments in algebraic geometry (broadly construed) over p-adically complete commutative rings. These developments include foundational advances
Closure-theoretic proofs of uniform bounds on symbolic powers in regular rings
. We give short, closure-theoretic proofs for uniform bounds on the growth of symbolic powers of ideals in regular rings. The author recently proved these bounds in mixed characteristic using a new
Coherence of absolute integral closures
. We prove that the absolute integral closure R + of an equicharacteristic zero noetherian complete local domain R is not coherent, provided dim( R ) ≥ 2. As a corollary, we give an elementary proof
Gluing for stable families of surfaces in mixed characteristic
Motivated by the question of properness of the moduli space of stable surfaces in mixed characteristic, we study the gluing problem for stable families of lc surfaces over a mixed characteristic DVR.
Mori Fibrations in Mixed Characteristic
This paper resolves several outstanding questions regarding the Minimal Model Program for klt threefolds in mixed characteristic. Namely termination for pairs which are not pseudo-effective,
Relative vanishing theorems for $\mathbf{Q}$-schemes
We prove the relative Grauert–Riemenschneider vanishing, Kawamata–Viehweg vanishing, and Kollár injectivity theorems for excellent Q-schemes, solving conjectures of Boutot and Kawakita. Our proof


Semistable Minimal Models of Threefolds in Positive or Mixed Characteristic
We extend the minimal model theorem to the 3-dimensional schemes which are projective and have semistable reduction over the spectrum of a Dedekind ring.
Generic vanishing in characteristic p > 0 and the characterization of ordinary abelian varieties
We prove a generic vanishing type statement in positive characteristic and apply it to prove positive characteristic versions of Kawamata's theorems: a characterization of smooth varieties birational
Minimal model program for log canonical threefolds in positive characteristic
Given a three-dimensional projective log canonical pair over a perfect field of characteristic larger than five, there exists a minimal model program that terminates after finitely many steps.
On the relative Minimal Model Program for fourfolds in positive characteristic
We show the validity of two special cases of the four dimensional Minimal Model Program in characteristic $p>5$: for contractions to $\mathbb{Q}$-factorial fourfolds and in families over curves. Our
Rational points on log Fano threefolds over a finite field
We prove the $W\mathcal{O}$-rationality of klt threefolds and the rational chain connectedness of klt Fano threefolds over a perfect field of characteristic $p>5$. As a consequence, any klt Fano
Perfectoid Spaces
We introduce a certain class of so-called perfectoid rings and spaces, which give a natural framework for Faltings’ almost purity theorem, and for which there is a natural tilting operation which