Related papers: On base point freeness in positive characteristic

Motivated by the change-of-domain problem for additive bases, Bukh, van Hintum and Keevash conjectured that if \(A,B\subseteq \mathbb{Q}^{n}\) and \(\{\boldsymbol{e}_i+\boldsymbol{e}_j:1\le i\le j\le n\}\subseteq A+B,\) then \(|A|+|B|\ge…

We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially larger than |A| for an arbitrary polynomial f over F_p. Second, a characterization is…

In the current paper we show that the dimension of a family $V$ of irreducible reduced curves in a given ample linear system on a toric surface $S$ over an algebraically closed field is bounded from above by $-K_S.C+p_g(C)-1$, where $C$…

Let $P: \F \times \F \to \F$ be a polynomial of bounded degree over a finite field $\F$ of large characteristic. In this paper we establish the following dichotomy: either $P$ is a moderate asymmetric expander in the sense that $|P(A,B)|…

We prove lifting theorems for completely positive maps going out of exact $C^\ast$-algebras, where we remain in control of which ideals are mapped into which. A consequence is, that if $\mathsf X$ is a second countable topological space,…

Let $(X, \Delta)$ be a projective klt pair of dimension $2$ and let $L$ be a nef $\mathbb{Q}$-divisor on $X$ such that $K_X + \Delta + L$ is nef. As a complement to the Generalized Abundance Conjecture by Lazi\'c and Peternell, we prove…

Let $(X,B)$ be a complex projective klt pair, and let $f\colon X\to Z$ be a surjective morphism onto a normal projective variety with maximal albanese dimension such that $K_X+B$ is relatively big over $Z$. We show that such pairs have good…

We prove that certain vector bundles over surfaces are ample if they are so when restricted to divisors, certain numerical criteria hold, and they are semistable (with respect to $\det(E)$). This result is a higher-rank version of a theorem…

We consider the problem of defining polynomials over function fields of positive characteristic. Among other results, we show that the following assertions are true. 1. Let $\G_p$ be an algebraic extension of a field of $p$ elements and…

We show that if X is a nonsingular projective variety of general type over an algebraically closed field k of positive characteristic and X has maximal Albanese dimension and the Albanese map is separable, then |4K_X| induces a birational…

The purpose of this paper is to translate positivity properties of the tangent bundle (and the anti-canonical bundle) of an algebraic manifold into existence and movability properties of rational curves and to investigate the impact on the…

Let $L$ be an ample line bundle over a smooth projective toric surface $X$. Then $L$ corresponds to a very ample lattice polytope $P$ that encodes many geometric properties of $L$. In this article, by studying $P$, we will give some…

For any affine hypersurface defined by a complete symmetric polynomial in $k\geq 3$ variables of degree $m$ over the finite field $\mathbb{F}_{q}$ of $q$ elements, a special case of our theorem says that this hypersurface has at least…

Let $\mathbb{K}$ be an algebraically closed field of characteristic $p>5$. We show the existence of minimal models for pseudo-effective NQC lc generalized pairs in dimension three over $\mathbb{K}$. As a consequence, we prove the…

We prove the base point free theorem for big line bundles on a three-dimensional log canonical projective pair defined over the algebraic closure of a finite field.

Let $(X,\Delta)$ be a proper dlt pair and $L$ a nef Cartier divisor such that $aL-(K_X+\Delta)$ is nef and log big on $(X,\Delta)$ for some $a\in {\mathbb Z}_{>0}$. Then $|mL|$ is base point free for every $m\gg 0$.

Let $K$ be a field of characteristic $0$, and let $k \geq 2$ be an integer. We prove that every $K$-linear bijection $f \colon K[X] \to K[X]$ strongly preserving the set of $k$-free polynomials (or the set of polynomials with a $k$-fold…

We give algebraic conditions about a finite algebra $B$ over a perfect field of positive characteristic, which are equivalent to the companionability of the theory of fields with "$B$-operators" (i.e. the operators coming from homomorphisms…

If $C$ is a curve over $\mathbb{Q}$ with genus at least $2$ and $C(\mathbb{Q})$ is empty, then the class of fields $K$ of characteristic 0 such that $C(K) = \varnothing$ has a model companion, which we call $C\mathrm{XF}$. The theory…

We prove that a general Fano hypersurface in a projective space over an algebraically closed field of arbitrary characteristic is separably rationally connected.