Related papers: A base point free theorem for log canonical surfac…

We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this theorem, the first one is to show an analogue of the…

Under some conditions on the deformation type, which we expect to be satisfied for arbitrary irreducible symplectic varieties, we describe which big and nef line bundles on irreducible symplectic varieties have base divisors. In particular,…

We determine the fixed locus of the anticanonical complete linear system of a given anticanonical rational surface. The case of a geometrically ruled rational surface is fully studied, e.g., the monoid of numerically effective divisor…

We establish the minimal model theory for $\mathbb Q$-factorial log surfaces and log canonical surfaces in Fujiki's class $\mathcal C$.

We discuss the cone theorem for quasi-log schemes and the Mori hyperbolicity. In particular, we establish that the log canonical divisor of a Mori hyperbolic projective normal pair is nef if it is nef when restricted to the non-lc locus.…

Let $X$ be a smooth algebraic curve. Suppose that there exists a triple covering $f : X \to Y$ where $Y$ is a smooth algebraic curve. In this paper, we investigate the existence of morphisms from $X$ to the projective line $\mathbf{P}^1$…

I give a necessary and sufficient condition for a nef and big line bundle in positive characteristic to be semi-ample, and then give two applications: I show that the relative dualizing sheaf of the universal curve is semi-ample, in…

More strong version of the main inductive theorem about the complements on surfaces is proved and the models of exceptional log del Pezzo surfaces with $\delta=0$ are constructed

We prove that the Cox ring of a smooth rational surface with big anticanonical class is finitely generated. We classify surfaces of this type that are blow-ups of the plane at distinct points lying on a (possibly reducible) cubic.

The purpose of this article is to develop techniques for estimating basis log canonical thresholds on logarithmic surfaces. To that end, we develop new local intersection estimates that imply log canonicity. Our main motivation and…

We show that the Bogomolov-Sommese vanishing theorem holds for a log canonical projective surface in large characteristic unless the Iitaka dimension of the round-down of the log canonical divisor is not equal to two. As an application, we…

For a smooth germ of algebraic variety $(X,0)$ and a hypersurface $(f=0)$ in $X$, with an isolated singularity at $0$, Teissier conjectured a lower bound for the Arnold exponent of $f$ in terms of the Arnold exponent of a hyperplane section…

We consider the Zariski-Lipman Conjecture on free module of derivations for algebraic surfaces. Using the theory of non-complete algebraic surfaces, and some basic results about ruled surfaces, we will prove the conjecture for several…

Let $C$ be a smooth plane curve of degree $d$ defined over an algebraically closed field $k$. A base point free complete very special linear system $g^r_n$ on $C$ is trivial if there exists an integer $m\ge 0$ and an effective divisor $E$…

In this paper we give an elementary proof of the Zariski-Lipman conjecture for log canonical spaces.

We discuss the relative log minimal model theory for log surfaces in the analytic setting. More precisely, we show that the minimal model program, the abundance theorem, and the finite generation of log canonical rings hold for log pairs of…

We prove that a projective vertical exact log smooth morphism of fs log analytic spaces with a base of log rank one yields polarized log Hodge structures in the canonical way.

In this paper, we prove the abundance theorem for numerically trivial canonical divisors on strongly $F$-regular varieties, assuming that the geometric generic fibers of the Albanese morphisms are strongly $F$-regular.

We prove an interpolation theorem for bounded free holomorphic functions.

We prove sharp $L^2$ Fourier restriction inequalities for compact, smooth surfaces in $\mathbb{R}^3$ equipped with the affine surface measure or a power thereof. The results are valid for all smooth surfaces and the bounds are uniform for…