Related papers: Nonspecial varieties and Generalized Lang-Vojta co…

Extending Culler-Shalen theory, Hara and the second author presented a way to construct certain kinds of branched surfaces in a $3$-manifold from an ideal point of a curve in the $\operatorname{SL}_n$-character variety. There exists an…

Let $X$ be a smooth projective variety over a number field $k$. The Green--Griffiths--Lang conjecture relates the question of finiteness of rational points in $X$ to the triviality of rational maps from abelian varieties to $X$ and to…

In this paper we prove that given a pair $(X,D)$ of a threefold $X$ and a boundary divisor $D$ with mild singularities, if $(K_X+D)$ is movable, then the orbifold second Chern class $c_2$ of $(X,D)$ is pseudo-effective. This generalizes the…

We pursue various restricted variable generalizations of the Chevalley-Warning theorem for low degree polynomial systems over a finite field. Our first such result involves variables restricted to Cartesian products of the Vandermonde…

The Weak Gravity Conjecture imposes stringent constraints on effective field theories to allow for an ultraviolet completion within quantum gravity. While substantial evidence supports the conjecture across broad classes of string…

In recent decades, linear affine threefolds have enabled researchers to solve some of the challenging problems on affine spaces. Koras-Russell threefolds, especially the Russell Cubic over $\mathbb{C}$ and Asanuma threefolds over a field of…

We show that infinitely many Gorenstein weakly-exceptional quotient singularities exist in all dimensions, we prove a weak-exceptionality criterion for five-dimensional quotient singularities, and we find a sufficient condition for being…

We prove that almost all random subsets of a finite vector space are weak Salem sets (small Fourier coefficient), which extends a result of Hayes to a different probability model.

This is the part II of our series of two papers, "Clemens' conjecture: part I", "Clemens' conjecture: part II". Continuing from part I, in this paper we turn our attention to general quintic threefolds. In a universal quintic threefold X,…

We give a counterexample to the Arithmetic Puncturing Conjecture and Geometric Puncturing Conjecture of Hassett-Tschinkel using symmetric powers of uniruled surfaces, and propose a corrected conjecture inspired by Campana's conjectures on…

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…

We construct smooth families of elliptic surface pairs with terminal singularities over a DVR of positive or mixed characteristic $(X,B)\to \mathrm{Spec}R$, such that $P_m(X_k,B_k)>P_m(X_K,B_K)$ for all sufficiently divisible $m>0$. In…

This is Part II of a series of three papers. We studies the hyperbolicity of complex quasi-projective varieties $X$ in the presence of a big and reductive representation $\varrho: \pi_1(X)\to {\rm GL}_N(\mathbb{C})$. For any Galois…

Let $X$ be a smooth projective variety over a finite field $\F$. We discuss the unramified cohomology group $H^3_\nr(X,\Q/\Z(2))$. Several conjectures put together imply that this group is finite. For certain classes of threefolds,…

Let $f : X \to S$ be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base $S$, and let $\mathbb{V} = R^{2k} f_{*} \mathbb{Z}(k)$ be the integral variation of Hodge structure coming from degree $2k$…

The minimal Weak Gravity Conjecture (WGC) predicts the emergence of towers of superextremal states in both weak and strong coupling limits. In this work, we study M-theory compactified on a special class of Calabi-Yau threefolds to…

A deep conjecture on torsion anomalous varieties states that if $V$ is a weak-transverse variety in an abelian variety, then the complement $V^{ta}$ of all $V$-torsion anomalous varieties is open and dense in $V$. We prove some cases of…

Let $A$ be a finite-dimensional algebra over an algebraically closed field. We prove $A$ is a strongly derived unbounded algebra if and only if there exists an integer $m$, such that $C_m(\proj A)$, the category of all minimal projective…

Over a field $k$, we study rational UFDs of finite transcendence degree $n$ over $k$. We classify such UFDs $B$ when $n=2$, $k$ is algebraically closed, and $B$ admits a positive $\mathbb{Z}$-grading, showing in particular that $B$ is…

A classification theorem is given of projective threefolds that are covered by a two-dimensional family of lines, but not by a higher dimensional family.