相关论文: Enumerating curves on rational surfaces: the ratio…
We answer some enumerative questions about irreducible rational curves on Hirzebruch surfaces, by combining an idea of Kontsevich with the study of the geometry of certain natural parameter spaces. Our formulas generalize Kontsevich's…
This is a survey describing recents developments in enumerative geometry of curves on projective varieties. Various methods to arrive at results such as Kontsevich's formula for plane rational curves, or Caporaso-Harris's formula for plane…
We analyze morphisms from pointed curves to K3 surfaces with a distinguished rational curve, such that the marked points are taken to the rational curve, perhaps with specified cross ratios. This builds on work of Mukai and others…
Kontsevich's formula for rational plane curves is a recursive relation for the number $N_d$ of degree $d$ rational curves in $\mathbb{P}^2$ passing through $3d-1$ general points. We provide two proofs of this recursion: the first more…
We present an approach to a large class of enumerative problems concerning rational curves in projective spaces. This approach uses analysis to obtain topological information about moduli spaces of stable maps. We demonstrate it by…
Various methods have been used to construct rational points and rational curves on rationally connected algebraic varieties. We survey recent advances in two of them, the descent and the fibration method, in a number-theoretical context…
We provide a real analog of the Yau-Zaslow formula counting rational curves on $K3$ surfaces.
Tyomkin's correspondence theorem states the equality of counts of rational curves of fixed homology class in a toric surface satisfying point and cross-ratio conditions with their tropical counterparts. Such correspondence theorems allow us…
We study the family of rational curves on arbitrary smooth hypersurfaces of low degree using tools from analytic number theory.
In characteristic $p>0$ and for $q$ a power of $p$, we compute the number of nonplanar rational curves of arbitrary degrees on a smooth Hermitian surface of degree $q+1$ under the assumption that the curves have a parametrization given by…
We obtain an explicit formula for the number of rational cuspidal curves of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as an Euler…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
We prove that some Gromov-Witten numbers associated to rational contact (Legendrian) curves in any contact complex projective space with arbitrary incidence conditions are enumerative. Also, we use Bott formula on the Kontsevich space to…
We investigate the existence, and lack of unicity, of a holomorphic fibration by discs transversal to a rational curve in a complex surface.
The aim of these notes is to explain the remarkable formula found by Yau and Zaslow to express the number of rational curves on a K3 surface. Projective K3 surfaces fall into countably many families F(g) (g>0); a surface in F(g) admits a…
We continue the development of methods for enumerating nodal curves on smooth complex surfaces, stressing the range of validity. We illustrate the new methods in three important examples. First, for up to eight nodes, we confirm…
By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface, a projective homogeneous…
We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite…
In this survey we discuss the problem of the existence of rational curves on complex surfaces, both in the K\"ahler and non-K\"ahler setup. We systematically go through the Enriques--Kodaira classification of complex surfaces to highlight…