Related papers: Globally +-regular varieties and the minimal model…
In this paper, we study the minimal model theory for threefolds in mixed characteristic. As a generalization of a result of Kawamata, we show that the MMP holds for strictly semi-stable schemes over an excellent Dedekind scheme $V$ of…
We show the validity of the Minimal Model Program for threefolds in characteristic five.
We provide a detailed proof of the validity of the Minimal Model Program for threefolds over excellent Dedekind separated schemes whose residue fields do not have characteristic 2 or 3.
We show the validity of the relative dlt MMP over Q-factorial threefolds in all characteristics p>0. As a corollary, we generalise many recent results to low characteristics including: $W\mathcal{O}$-rationality of klt singularities,…
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, finiteness of minimal models and the Sarkisov…
We prove that globally $+$-regular varieties are rationally chain connected in dimension three and mixed characteristic with residue field characteristic $p>5$. We also introduce a notion of strongly globally $+$-regular, and show that…
We show the abundance theorem for arithmetic klt threefold pairs whose closed point have residue characteristic greater than five. As a consequence, we give a sufficient condition for the asymptotic invariance of plurigenera for certain…
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 ("semi-stable mmp"). We also provide their…
Let $f:(X,B)\to Z$ be a 3-fold extremal dlt flipping contraction defined over an algebraically closed field of characteristic $p>5$, such that the coefficients of $\{B\}$ are in the standard set $\{1-\frac 1n|n\in \mathbb N\}$, then the…
Let $R = W(k)$ be the ring of Witt vectors over an algebraically closed field $k$ of characteristic $p > 2$. Let $M$ be a three-dimensional regular integral flat projective $R$-scheme such that $H^0(M,\mathcal{O}_M) = R$ and the…
The regular models of a normal logic program are a particular type of partial (i.e. 3-valued) models which correspond to stable partial models with minimal undefinedness. In this paper, we explore graphical conditions on the dependency…
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.
We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of…
This is the second of a series of papers studying real algebraic threefolds using the minimal model program. The main result is the following. Let $X$ be a smooth projective real algebraic 3-fold. Assume that the set of real points is an…
If $(X, \mcF, \D)$ is a projective rank two foliated log canonical triple such that $(X,B)$ is klt for some $0 \leq B \leq \D$, we show that we can run a $(K_\mcF +\Delta)$-MMP and any such MMP terminates with either a minimal model or Mori…
Nakayama showed that deformation invariance of plurigenera for smooth complex varieties follows from the MMP and Abundance Conjectures. We generalize his result to families of singular pairs over DVRs of positive or mixed characteristic. As…
Suppose that $X$ is an integral scheme (quasi-)projective over a complete local ring of mixed characteristic. Using ideas of Takamatsu-Yoshikawa and Bhatt-Ma-et. al, we define a notion of a $+$-test ideal on $X$, including for divisors and…
An estimate for the genus function in circle bundles over irreducible 3-manifolds is proven. This estimate is in many cases an equality and it relates the minimal genus of the surfaces representing a given homology class with the…
This is the first of a series of papers studying real algebraic threefolds using the minimal model program. The main results are outlined in Part II. The present part I. contains the necessary preliminary work concerning terminal…
Using techniques from the theory of foliations, we establish the cone theorem and the contraction theorem for lc generalized pairs in full generality, and meanwhile develop the minimal model program for $\mathbb Q$-factorial foliated dlt…