Let $S$ be KLT threefold singularity over an algebraically closed field of positive characteristic $p>5$. We prove that its local étale fundamental group is tame and finite. Further, we show that every finite unipotent torsor over a big open of $S$ is realized as the restriction of a finite unipotent torsor over $S$.
In this article we present a refinement of the base point free theorem for threefolds in positive characteristic. If $L$ is a nef Cartier divisor of numerical dimension at least one on a projective Kawamata log terminal threefold $(X,Δ)$ over a perfect field $k$ of characteristic $p \gg 0$ such that $L-(K_X+Δ)$ is big and nef, then we show that the linear system $|mL|$ is base point free for all sufficiently large integer $m>0$.
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.
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.
Let $(X,Δ)$ be an $n$-dimensional $ε$-klt log $\QQ$-Fano pair. We give an upper bound for the volume ${\rm Vol}(-(K_X+Δ))=(-(K_X+Δ))^n$ when $n=2$ or $n=3$ and $X$ is {$\QQ$-factorial} of $ρ(X)=1$. This bound is essentially sharp for $n=2$. Existence of an upper bound for anticanonical volumes is related the Borisov-Alexeev-Borisov Conjecture which asserts boundedness of the set of $ε$-klt log $\QQ$-Fano varieties of a given dimension $n$.
We prove an analogue of Fujino and Mori's ``bounding the denominators'' in the log canonical bundle formula (see also Prokhorov and Shokurov) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a klt pair $(X,Δ)$ of Kodaira codimension one and dimension at most three such that the coefficients of $Δ$ are in a DCC set $\mathcal{A}$, there is a natural number $N$ that depends only on $\mathcal{A}$ for which the round down of $\N(K_X+Δ)$ induces the Iitaka fibration. We also prove a birational boundedness result for klt surfaces of general type.
In this paper, we establish a vanishing theorem of Nadel type for the Witt multiplier ideals on threefolds over perfect fields of characteristic larger than five. As an application, if a projective normal threefold over $\mathbb{F}_q$ is not klt and its canonical divisor is anti-ample, then the number of the rational points on the klt-locus is divisible by $q$.
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 families of singular surface pairs to hold in mixed characteristic.
Let $(X, Δ)$ be a projective klt three dimensional pair defined over an algebraically closed field characteristic larger than 5. Let $L$ be a nef and big line bundle on $X$ such that $L-K_X-Δ$ is big and nef. We show that $L$ is indeed semi-ample.
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, inversion of adjunction, and normality of divisorial centres up to a universal homeomorphism.
Let $X$ be a normal projective threefold with mild singularities, and $L_X$ a strictly nef $\mathbb{Q}$-divisor on $X$. First, we show the ampleness of $K_X+tL_X$ with sufficiently large $t$ if either the Kodaira dimension $κ(X)\neq 0$ or the augmented irregularity $q^{\circ}(X)\neq 0$. Second, we show that, if $(X,Δ)$ is a projective klt threefold pair with the anti-log canonical divisor $-(K_X+Δ)$ being strictly nef, then $X$ is rationally connected.
In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$.
In this article we study conjectures regarding normalized volume and boundedness of singularities. We focus on singularities with a torus action of complexity 1, threefold singularities, and hypersurface singularities. Given a real value v>0, we prove that the class of K-semistable threefold singularities with normalized volume at least v forms a bounded family. Analogous statements are proved in the case of n-dimensional complexity-1 and n-dimensional hypersurface singularities for arbitary n. In the general case of klt singularities, i.e. without the assumption on K-semistability, we show th…
We show that a weak version of the canonical bundle formula holds for fibrations of relative dimension one. We provide various applications thereof, for instance, using the recent result of Xu and Zhang, we prove the log non-vanishing conjecture for three-dimensional klt pairs over any algebraically closed field $k$ of characteristic $p>5$. We also show the log abundance conjecture for threefolds over $k$ when the nef dimension is not maximal, and the base point free theorem for threefolds over the algebraic closure of any finite field of characteristic $p>2$.
We study compactifications of Drinfeld half-spaces over a finite field. In particular, we construct a purely inseparable endomorphism of Drinfeld's half-space $Ω(V)$ over a finite field $k$ that does not extend to an endomorphism of the projective space $P (V)$. This should be compared with theorem of Rémy, Thuillier and Werner that every $k$-automorphism of $Ω(V)$ extends to a $k$-automorphism of $P (V)$. Our construction uses an inseparable analogue of the Cremona transformation. We also study foliations on Drinfeld's half-spaces. This leads to various examples of interesting varieties in po…
