Related papers: Fibration by non-smooth projective curves of arith…

We give a complete classification, up to birational equivalence, of all fibrations by plane projective rational quartic curves in characteristic two.

In 1944 Zariski discovered that Bertini's theorem on variable singular points is no longer true when we pass from a field of characteristic zero to a field of positive characteristic. In other words, he found fibrations by singular curves,…

We classify fibrations by integral plane projective rational quartic curves whose generic fibre is regular but admits a non-smooth point that is a canonical divisor. These fibrations can only exist in characteristic two. The geometric…

In this paper we classify curves of genus two over a perfect field k of characteristic two. We find rational models of curves with a given arithmetic structure for the ramification divisor and we give necessary and sufficient conditions for…

We prove Bertini type theorems and give some applications of them. The applications are in the context of Lefschetz theorem for Nori fundamental group for normal varieties as well as for geometric formal orbifolds. In another application,…

We investigate fibrations by non-hyperelliptic curves of arithmetic genus three and geometric genus one in characteristic two. Assuming that there is only one moving singularity and that its image in the Frobenius pullback of the fibration…

We determine the central simple algebras D over a functionfield K of trancendence degree two which admit a model of smooth Cayley-Hamilton algebras. This happens if and only if there is a smooth model S of K such that the ramification…

In this paper we explore conditions for a curve in a smooth projective surface to have a free product of cyclic groups as the fundamental group of its complement. It is known that if the surface is $\mathbb P^2$, then such curves must be of…

Let $A$ be an abelian variety and $G$ a finite group of automorphisms of $A$ fixing the origin such that $A/G$ is smooth. The quotient $A/G$ can be seen as a fibration over an abelian variety whose fibers are isomorphic to a product of…

We study deformation theory of elliptic fibre bundles over curves in positive characteristics. As applications, we give examples of non-liftable elliptic surfaces in charactertic two and three, which answers a question of Katsura and Ueno.…

Let $\mathcal{X}\rightarrow C$ be a dominant morphism between smooth irreducible varieties over a finitely generated field $k$ such that the generic fiber $X$ is smooth, projective and geometrically connected. Assuming that $C$ is a curve…

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…

We consider families of smooth projective curves of genus 2 with a single point removed and study their integral points. We show that in many such families there is a dense set of fibres for which the integral points can be effectively…

We study stable curves of arithmetic genus 2 which admit two morphisms of finite degree $p$, resp. $d$, onto smooth elliptic curves, with particular attention to the case $p$ prime.

Using the theory of moduli of curves, we establish various slope inequalities for general fibered surfaces. More precisely, we introduce the notion of functorial divisors on Artin stacks and prove a theorem concerning their effectiveness.…

We consider relatively minimal fibrations of curves of genus two on rational surfaces whose Picard numbers are not maximal. By birational morphisms, such fibred surfaces are interpreted as pencils of plane curves. We show that only four are…

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…

We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves $B$ in smooth projective surfaces $X$ defined over an algebraically closed field of characteristic zero. We observe that…

Consider the scheme parametrizing non-constant morphisms from a fixed projective curve to a projective surface. There is a rational map between this scheme and the Chow variety of $1$-cycles on the surface. We prove that, if the curve is…

We give an algorithm to decide if a given braid is a product of two factors which are conjugates of given powers of standard generators of the braid group. The same problem is solved in a certain class of Garside groups including Artin-Tits…