Related papers: Rational Singularities and Rational Points

Let $X$ be a projective variety with log terminal singularities and vanishing augmented irregularity. In this paper we prove that if $X$ admits a relatively minimal genus one fibration then it does contain a subvariety of codimension one…

We undertake a study of conic bundle threefolds $\pi\colon X\to W$ over geometrically rational surfaces whose associated discriminant covers $\tilde{\Delta}\to\Delta\subset W$ are smooth and geometrically irreducible. First, we determine…

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…

Recently Fukasawa, Homma and Kim introduced and studied certain projective singular curves over $\mathbb {F}_q$ with many extremal properties. Here we extend their definition to more general non-rational curves.

We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal…

Using the construction of Colliot-Th\'el\`ene and Ojanguren, we exhibit an example of a smooth projective geometrically rational variety X defined over a finite field F_p with an algebraic closure \bar F_p and the absolute Galois group G,…

Let $\mathcal{E}_{f}:y^2=x^3+f(t)x$, where $f\in\Q[t]\setminus\Q$, and let us assume that $\op{deg}f\leq 4$. In this paper we prove that if $\op{deg}f\leq 3$, then there exists a rational base change $t\mapsto\phi(t)$ such that on the…

We study the structure of the rational motivic stable homotopy category over general base schemes. Our first class of results concerns the six operations: we prove absolute purity, stability of constructible objects, and…

In this paper we give a necessary combinatorial condition for a negative--definite plumbing tree to be suitable for rational blow--down, or to be the graph of a complex surface singularity which admits a rational homology disk smoothing.…

In this paper certain Chow weight structures on the "big" triangulated motivic categories $DM_R^{eff}\subset DM_R$ are defined in terms of motives of all smooth varieties over the base field. This definition allows studying basic properties…

The families of smooth rational surfaces in $\PP^4$ have been classified in degree $\le 10$. All known rational surfaces in $\PP^4$ can be represented as blow-ups of the plane $\PP^2$. The fine classification of these surfaces consists of…

A central problem in Diophantine geometry is to uniformly bound the number of $K$-rational points on a smooth curve $X/K$ in terms of $K$ and its genus $g$. A recent paper by Stoll proved uniform bounds for the number of $K$-rational points…

The topology of the orbit space, $Y$, for the action of the complex conjugation on a complex surface, $X$, defined over reals, is studied. I give a criterion for blow-up stable triviality of $Y$ (which implies vanishing of its…

We effectively bound T-singularities on non-rational projective surfaces with an arbitrary amount of T-singularities and ample canonical class. This fully generalizes the previous work for the case of one singularity, and illustrates the…

Let X be a normal variety such that $K_X$ is Q-Cartier, and let $f: X \rightarrow X$ be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that…

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.

Let $X$ be a smooth irreducible projective curve of genus $g \geq 2$ over a finite field $\F_{q}$ of characteristic $p$ with $q$ elements such that the function field $\F_{q}(X)$ is a geometric Galois extension of the rational function…

We prove using jet schemes that the zero loci of the moment maps for the quivers with one vertex and at least two loops have rational singularities. This implies that the spaces of representations of the fundamental group of a compact…

It is well known that a smooth projective Fano variety is rationally connected. Recently Zhang (and later Hacon and McKernan as a special case of their work on the Shokurov RC-conjecture) proved that the same conclusion holds for a klt pair…

Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about a rational point if a bound is…