Related papers: Negative curves on algebraic surfaces

Let $X$ be a rational surface obtained by blowing up at a configuration $\mathcal{C}$ of infinitely near points over a Hirzebruch surface $\mathbb{F}_\delta$. We prove that there exist two positive integers $a \leq b$ such that the cone of…

Esnault-Viehweg developed the theory of cyclic branched coverings $\tilde X\to X$ of smooth surfaces providing a very explicit formula for the decomposition of $H^1(\tilde X,\mathbb{C})$ in terms of a resolution of the ramification locus.…

In this paper, we study the deformations of curves in the projective 3-space $\mathbb P^3$ (space curves), one of the most classically studied objects in algebraic geometry. We prove a conjecture due to J. O. Kleppe (in fact, a version…

We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer…

This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra…

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic zero, prove almost all cases in positive characteristic and improve the proofs of the previously…

This paper concerns the question of whether a more direct limit can be used to obtain the limit Hilbert-Kunz multiplicity, a possible candidate for a characteristic zero Hilbert-Kunz multiplicity. The main goal is to establish an…

In this note we address the following kind of question: let X be a smooth, irreducible, projective surface and D a divisor on X$satisfying some sort of positivity hypothesis, then is there some multiple of D depending only on X which is…

I construct "fake algebraic curves" in $Cp^2$. More precisely, for any k>2, I construct infinitely many pairwise smoothly non-isotopic (and moreover not ambient diffeomorphic) smooth surfaces $F\subset Cp^2$ homeomorphic to a non-singular…

We study points of moderately low degree on a curve $C$ over a number field, which is embedded on a nice toric surface $S$. Recently, Smith and Vogt related the linear equivalence classes of such points to intersections of $C$ with curves…

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…

There exists a dictionary between hereditary orders and smooth stacky curves, resp. tame orders of global dimension 2 and Azumaya algebras on smooth stacky surfaces. We extend this dictionary by explaining how the restriction of a tame…

Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…

We describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction. These examples provide evidence for the Eichler-Shimura conjecture for Hilbert modular forms…

We use Morse theoretical arguments to study algebraic curves in C^2. We take an algebraic curve C in C^2 and intersect it with a family of spheres with fixed origin and varying radii. We explain in detail how does the resulting link change…

The paper contains a general construction which produces new examples of non simply-connected smooth projective surfaces. We analyze the resulting surfaces and their fundamental groups. Many of these fundamental groups are expected to be…

Properties of the recently reported homogeneous Hilbert curves are deduced and reported. The nature of the affine transformations involved in the construction of the Hilbert curves is explored. The analytical representation of proper and…

Suppose B=F[x,y,z]/h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve C with a node. Let J be a homogeneous (x,y,z)-primary ideal and n -> e_n be the Hilbert-Kunz function of B with respect to J. Let…

We study the number of rational points of smooth projective curves over finite fields in some relative situations in the spirit of a previous paper from an euclidean point of vue. We prove some kinds of relative Weil bounds, derived from…

For a smooth projective variety $X$ over a number field $k$ a conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map of $X$ is a torsion group. In this article we consider a product $X=C_1\times\cdots\times C_d$ of…