Related papers: Rational self-maps of projective surfaces with a r…

It is proved that on a smooth algebraic variety, fibered into cubic surfaces over the projective line and sufficiently ``twisted'' over the base, there is only one pencil of rational surfaces -- that is, this very pencil of cubics. In…

Let $S$ be the collection of quadratic polynomial maps, and degree $2$-rational maps whose automorphism groups are isomorphic to $C_2$ defined over the rational field. Assuming standard conjectures of Poonen and Manes on the period length…

There are many examples of 3-folds of general type with $\chi(\mathcal {O})=1$ found by Iano-Fletcher and Reid about twenty years ago. Iano-Fletcher has ever proved $P_{12}(X)\ge 1$ and $P_{24}(X)\ge 2$ for all minimal 3-folds $X$ of…

A rational map whose source and image are projectively embedded varieties has an {\em Arithmetically Cohen-Macaulay graph} if the Rees algebra of one (hence any) of its base ideals is a Cohen-Macaulay ring. If the map is birational onto the…

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices over a field $\mathbb{F}$ of characteristic not equal to $2$. If $n\ge 2$, we show that an arbitrary map $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$ is Jordan multiplicative,…

A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven…

The number of rational points of a plane non-singular algebraic curve X defined over a finite field is computed, provided that the generic point of X is not an inflexion and that X is Frobenius non-classical with respect to conics.

To every automorphism w of an infinite rooted regular binary tree we associate a two variable generating function \Phi_w that encodes information on the orbit structure of w. We prove that this is a rational function if w can be described…

We continue to study birational geometry of Fano fibrations $\pi\colon V\to {\mathbb P}^1$ the fibers of which are Fano double hypersurfaces of index 1. For a majority of families of this type, which do not satisfy the condition of…

A tri-linear rational map in dimension three is a rational map $\phi: (\mathbb{P}_\mathbb{C}^1)^3 \dashrightarrow \mathbb{P}_\mathbb{C}^3$ defined by four tri-linear polynomials without a common factor. If $\phi$ admits an inverse rational…

The dynamical structure of the rational map $ax+1/x$ on the projective line over the field Q2 of $2$-adic numbers, is fully described.

We improve a bound due to the second author on number of rational points on smooth surfaces in $\mathbb{P}^3$ over finite fields and look at families of surfaces that achieve or nearly achieve this bound, for which we compute their exact…

Let $f \colon X \dashrightarrow X$ be a dominant rational self-map of a smooth projective variety defined over $\overline{\mathbb Q}$. For each point $P\in X(\overline{\mathbb Q})$ whose forward $f$-orbit is well-defined, Silverman…

We classify all complex surfaces with quotient singularities that do not contain any smooth rational curves, under the assumption that the canonical divisor of the surface is not pseudo-effective. As a corollary we show that if $X$ is a log…

We show that a real rational (over $\C$) surfaces are quasi-simple, i.e., that such a surface is determined up to deformation in the class of real surfaces by the topological type of its real structure.

For a graph $G$, a vertex coloring $f$ is called nonrepetitive if for all $k\in\mathbb N$ and all $P_{2k}=\langle v_1, \cdots, v_k,v_{k+1}, \cdots, v_{2k}\rangle$ (path of $2k$ vertices) in $G$, there must be some $1\le i\le k$ such that…

An outstanding folklore conjecture asserts that, for any prime $p$, up to isomorphism the projective plane $PG(2,\mathbb{F}_p)$ over the field $\mathbb{F}_p := \mathbb{Z}/p\mathbb{Z}$ is the unique projective plane of order $p$. Let $\pi$…

Given a sequence of bounded operators $a_j$ on a Hilbert space $H$ with $\sum a_j^*a_j=1=\sum a_ja_j^*$, we study the map $\Psi$ defined on $B(H)$ by $\Psi(x)=\sum a_j^*xa_j$ and its restriction $\Phi$ to the Hilbert-Schmidt class $C^2(H)$.…

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that…

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain $n \ge \b_2-2$ disjoint smooth rational curves with self-intersection -2, where $\b_2$ is the…