Searcharxiv⌕ Search

arXiv subjects

Hiromu Tanaka

Publications and source records attributed to Hiromu Tanaka.

At least 55 records · Page 3Linked to original sources
arXiv2018

Purely log terminal threefolds with non-normal centres in characteristic two

We show that many classical results of the minimal model programme do not hold over an algebraically closed field of characteristic two. Indeed, we construct a three dimensional plt pair whose codimension one part is not normal, a three dimensional klt singularity which is not rational nor Cohen-Macaulay, and a klt Fano threefold with non-trivial intermediate cohomology.

math.AG↗
arXiv2017

Semiample perturbations for log canonical varieties over an F-finite field containing an infinite perfect field

Let $k$ be an $F$-finite field containing an infinite perfect field of positive characteristic. Let $(X, Δ)$ be a projective log canonical pair over $k$. In this note we show that, for a semi-ample divisor $D$ on $X$, there exists an effective $\mathbb{Q}$-divisor $Δ' \sim_{\mathbb Q} Δ+D$ such that $(X, Δ')$ is log canonical if there exists a log resolution of $(X, Δ)$.

math.AG↗
arXiv2016

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 threefold over a finite field has a rational point.

math.AG↗
arXiv2016

On log del Pezzo surfaces in large characteristic

We show that any Kawamata log terminal del Pezzo surface over an algebraically closed field of large characteristic is globally F-regular or it admits a log resolution which is liftable to characteristic zero. As a consequence, we prove the Kawamata-Viehweg vanishing theorem for klt del Pezzo surfaces of large characteristic.

math.AG↗
arXiv2016

Behavior of canonical divisors under purely inseparable base changes

Let $k$ be an imperfect field. Let $X$ be a regular variety over $k$ and set $Y$ to be the normalization of $(X \times_k k^{1/p^{\infty}})_{\rm red}$. In this paper, we show that $K_Y+C=f^*K_X$ for some effective divisor $C$ on $Y$. We obtain the following three applications. First, we show that a $K_X$-trivial fiber space with non-normal fibers is uniruled. Second, we prove that general fibers of Mori fiber spaces are rationally chain connected. Third, we obtain a weakening of the cone theorem for surfaces and threefolds defined over an imperfect field.

math.AG↗
arXiv2015

Minimal models and abundance for positive characteristic log surfaces

We discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for Q-factorial surfaces and for log canonical surfaces. Moreover, in the case where the base field is the algebraic closure of a finite field, we obtain the same results under much weaker assumptions.

math.AG↗
arXiv2014

On base point freeness in positive characteristic

We prove that if $(X,A+B)$ is a pair defined over an algebraically closed field of positive characteristic such that $(X,B)$ is strongly $F$-regular, $A$ is ample and $K_X+A+B$ is strictly nef, then $K_X+A+B$ is ample. Similarly, we prove that for a log pair $(X,A+B)$ with $A$ being ample and $B$ effective, $K_X+A+B$ is big if it is nef and of maximal nef dimension. As an application, we establish a rationality theorem for the nef threshold and various results towards the minimal model program in dimension three in positive characteristic.

math.AG↗