Related papers: Rational curves and rational singularities
We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal…
A new generalization of the classical separate algebraicity theorem is suggested and proved.
In this paper we investigate complex uniruled varieties $X$ whose rational curves of minimal degree satisfy a special property. Namely, we assume that the tangent directions to such curves at a general point $x\in X$ form a linear subspace…
Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational 2-torsion point on their Jacobian.…
It is a conjecture of Koll\'ar that a variety $X$ with rational singularities in some open subvariety $U$ has a rationalification; that is, a proper, birational morphism $f: Y \rightarrow X$ such that $Y$ has rational singularities, and…
We study the "generic" degenerations of curves with two singular points when the points merge. First, the notion of generic degeneration is defined precisely. Then a method to classify the possible results of generic degenerations is…
This is an extended version of an invited lecture I gave at the Journees Arithmetiques in St. Etienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective)…
We study rational points on a smooth variety X over a complete local field K with algebraically closed residue field, and models of X with tame quotient singularities. If a model of X is the quotient of a Galois action on a weak N\'eron…
Based on ideas from recent joint work with Bjorn Poonen, we describe an algorithm that can in certain cases determine the set of rational points on a curve $C$, given only the $p$-Selmer group $S$ of its Jacobian (or some other abelian…
The study of algebraic curves $\cX$ with numerous automorphisms in relation to their genus $g(\cX)$ is a well-established area in Algebraic Geometry. In 1995, Irokawa and Sasaki \cite{Sasaki} gave a complete classification of curves over…
We consider flat families of reduced curves on a smooth surface S such that each member C has the same number of singularities of fixed singularity types and the corresponding (locally closed) subscheme H of the Hilbert scheme of S. We are…
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 prove that the incidence scheme of rational curves of degree 11 on quintic threefolds is irreducible. This implies a strong form of the Clemens conjecture in degree 11. Namely, on a general quintic threefold $F$ in $\mathbb{P}^4$, there…
We give a decomposition of the jacobian variety of a generalized Fermat curve. This extends a result obtained by Auffarth, Lucchini-Arteche and Rojas on Humbert-Edge curves, which are a particular case of generalized Fermat curves. (A…
We study the pull-back of regular 1-forms on a complex irreducible plane curve singularity under the normalization morphism.
In this article, we present a method for computing rational points on hyperelliptic curves of genus~3 and isolated quadratic points on hyperelliptic curves of genus~2 and~3 whose Jacobians have rank~0. Our approach begins by computing the…
In this article we investigate the natural domain of definition of a holonomy map associated to a singular holomorphic foliation of the complex projective plane. We prove that germs of holonomy between algebraic curves can have large sets…
A general canonical curve X determines a finite set T(X) of hyperplanes, which is in bijective correspondence with the set of odd theta-characteristics of X. The definition of T(X) can be extended to certain singular curves, in a way that…
We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves…
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…