Related papers: On Chisini's Conjecture. II

In this short note we prove a version of Bertini's theorem for unipotent rigid fundamental groups, stating that for every smooth, projective, geometrically connected variety $X$ over an infinite perfect field $k$ of characteristic $p>0$,…

We prove that if every chain on a strictly pseudoconvex hypersurface $M$ in $\mathbb{C}^2$ coincides with the boundary of a stationary disc, then $M$ is locally spherical.

We prove that, if two germs of plane curves $(C,0)$ and $(C',0)$ with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then $C$ is complex isomorphic to $C'$ or to $\overline{C'}$. A similar result was shown by…

A family of proper smooth curves of genus $\geq 2$, parametrised by an open dense subset $U$ of a normal variety $S$, extends to $S$ if the natural map $\pi_1(U) \to \pi_1(S)$ on fundamental groups is an isomorphism. The criterion of this…

The canonical degree $C.K_X$ of an integral curve on a smooth projective surface $X$ is conjecturally bounded from above by an expression of the form $A(g-1)+B$, where $g$ is the geometric genus of $C$ and $A$, $B$ are constants depending…

Throughout this paper we study the existence of irreducible curves C on smooth projective surfaces S with singular points of prescribed topological types S_1,...,S_r. There are necessary conditions for the existence of the type \sum_{i=1}^r…

The Severi variety V_{n,d} of a smooth projective surface S is defined as the subvariety of the linear system |O_S(n)|, which parametrizes curves with d nodes. We show that, for a general surface S of degree k in P^3 and for all n>k-1,…

We prove an analog of Belyi's theorem for the algebraic surfaces. Namely, any non-singular algebraic surface can be defined over a number field if and only it covers the complex projective plane with ramification at three knotted…

Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell-Weil theorem for V states that there is a finite subset P \subset V(k) such that the whole V(k) can be obtained from P by drawing secants and tangents…

We show that an abelian surface embedded in P^N by a very ample line bundle L of type (1,2d) is projectively normal if and only if d>=4. This completes the study of the projective normality of abelian surfaces embedded by complete linear…

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 consider a finite analytic morphism $\varphi =(f,g)$ defined from a complex analytic normal surface $(Z,z)$ to ${\mathbb C}^2$. We describe the topology of the image by $\varphi$ of a reduced curve on $(Z,z)$ by means of iterated pencils…

A Laurent polynomial $f$ in two variables naturally describes a projective curve $C(f)$ on a toric surface. We show that if $C(f)$ is a smooth curve of genus at least 7, then $C(f)$ is not Brill-Noether general. To accomplish this, we…

Let $f:S \fr B$ be a surface fibration with fibres of genus 5. We find a linear relation between the fundamental invariants of the surface. Namely $K_f^2=\chi_f+N$ where $N$ is the number of trigonal fibres. Our proof is based on the…

In this work, it is established that for a generic projective hypersurface $H\subset\mathbb{P}^n(\mathbb{C})$ of degree $d\geq(5n)^2\,n^{n}$, any holomorphic entire curve $f\colon\mathbb{C}\to\mathbb{P}^n(\mathbb{C})\setminus H$ has its…

We classify the minimal algebraic surfaces of general type with $p_g=q=1, K^2=8$ and bicanonical map of degree 2. It will turn out that they are isogenous to a product of curves, so that if $S$ is such a surface then there exist two smooth…

Let W -> A^2 be the universal Weierstrass family of cubic curves over C. For each N >= 2, we construct surfaces parametrizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to…

The main result in this paper is as follows: Let S be the branch curve (in the projective plan) of a generic projection of a Veronese surface. Then the fundamental group of the complement of S is an extension of a solvable group by a…

Segre proved that a smooth cubic surface over Q is unirational iff it has a rational point. We prove that the result also holds for cubic hypersurfaces over any field, including finite fields.

For a minimal smooth projective surface $S$ of general type over a field of characteristic $p>0$, we prove that $K^2_S\le 32\chi(\cal{O}_S).$ Moreover, if $18\chi(\cal{O}_S)<K^2_S\le 32\chi(\cal{O}_S)$, Albanese morphism of $S$ must induces…