Maycol Falla Luza, Frank Loray
We prove the existence of (non compact) complex surfaces with a smooth rational curve embedded such that there does not exist any formal singular foliation along the curve. In particular, at arbitray small neighborhood of the curve, any meromorphic function is constant. This implies that the Picard group is not countably generated.
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Neighborhood of a rational curve with an ordinary cusp
本文考察自交数为零的非奇异复曲面内带通常尖点的约化有理曲线，分析其邻域的复解析性质。
Arnol'd showed the uniqueness of the complex analytic structure of a small neighborhood of a non-singular elliptic curve embedded in a non-singular surface whose normal bundle satisfies Diophantine condition in the Picard variety. We show an analogue of this theorem for a neighborhood of a cycle of rational curves.
Stabilization of fields of meromorphic functions on neighborhoods of a rational curve
该研究针对含正自交光滑有理曲线的光滑连通复曲面，证明存在包含该曲线的邻域U，使得U内曲线连通邻域上的亚纯函数均可延拓至整个U，完善了复曲面亚纯函数延拓理论。
On fields of meromorphic functions on neighborhoods of rational curves
本文证明了包含正自交平滑有理曲线的复曲面上的亚纯函数域同构于 $\mathbb C$ 上的一元或二元有理函数域。
In this paper we study the formal classication of two-dimensional neighborhoods of genus g $\ge$ 2 curves with trivial normal bundle. We first construct formal foliations on such neighborhoods with holonomy vanishing along many loops, then give the formal/analytic classication of neighborhoods equipped with two foliations, and finally put this together to obtain a description of the space of neighborhoods up to formal equivalence.
Non-algebraizable neighborhoods of curves
本文提出了若干族嵌入光滑复曲面的紧复曲线，通过拼接射影曲面邻域和收缩例外曲线构造出无法嵌入代数曲面的邻域，推广了S. Lvovski的例子。
Geometry of Neighborhoods of Minimal Rational Curves
本文综述了单有理射影流形上极小有理曲线的芽等价问题，重点研究了当极小有理切线的相伴簇构成射影簇的等平凡族时，Zariski开子集上自然出现的G-结构及其与几何结构的微分几何的相互作用，并讨论了极小有理曲线芽等价的形式原理。
On algebraic and non-algebraic neighborhoods of rational curves
本文研究有理曲线邻域的代数性问题，构造了自交数任意正整数d的ℙ¹非代数嵌入芽，其亚纯函数芽域超越次数为2，并给出两种构造及基于ℙ¹嵌入代数芽分类的非代数性证明。
A previous result in a paper by Buch and Mihalcea relates the curve neighborhood of any Schubert variety to the curve neighborhood of a point and introduces the idea of P-cosmall roots. In our paper, we prove a conjecture given in that same paper about P-cosmall roots. We go on to provide explicit combinatorial conditions for a root to be P-cosmall.
Curve neighborhoods of Seidel products in quantum cohomology
本文针对Buch-Chaput-Perrin猜想，证明了在A型旗簇中，对应于Seidel型量子积的两点曲线邻域是一个显式给出的Schubert簇。
Let $\mbox{IG}(k,2n+1)$ be the odd symplectic Grassmannian. It is a quasi-ho\-mo\-ge\-neous space with homogeneous-like behavior. A very limited description of curve neighborhoods of Schubert varieties in $\mbox{IG}(k,2n+1)$ was used by Mihalcea and the second named author to prove an (equivariant) quantum Chevalley rule. In this paper we give a full description of the irreducible components of curve neighborhoods in terms of the Hecke product of (appropriate) Weyl group elements, $k$-strict partitions, and BC-partitions. The latter set of partitions respect the Bruhat order with inclusions. O…
Neighborhoods of curves with a prescribed number of foliations
本研究针对n维射影空间中连通射影曲线的邻域问题，构造了恰好带指定数目余维1全纯叶状结构的n维非紧复流形，确定其亚纯函数域的可能形式，推广并改进了曲面情形的相关结果。
We prove the existence of regular foliations with a prescribed tangency divisor in neighborhoods of negatively embedded holomorphic curves; this is related to a linearization theorem due to Grauert. We give also examples of neighborhoods which can not be linearized.
We investigate the existence, and lack of unicity, of a holomorphic fibration by discs transversal to a rational curve in a complex surface.
We use virtual neighborhood technique to establish GW-invariants, Quantum cohomology, equivariant GW-invariants, equivariant quantum cohomology and Floer cohomology for general symplectic manifold. We also establish GW-invariants for a family of symplectic manifolds. As a consequence, we prove Arnold conjecture for nondegenerate Hamiltonian symplectomorphisms.
Curve neighborhoods and combinatorial property $\mathcal{O}$ for a family of odd symplectic partial flag manifolds
该研究针对奇辛部分旗流形族$\u200b\mbox{IF}(1,2;E)$，完整刻画其任意Schubert簇的$d$次曲线邻域不可约分支、研究格结构，证明了猜想$\u0026#x1D4AA;$的组合版本。
Formalizing $A_1^{(1)}$ Curve Neighborhoods in Lean 4
本文在Lean 4中形式化了$A_1^{(1)}$类型的组合曲线邻域，通过无限二面体群$D_∞$的时刻图编码，并验证了任意元素的显式组合公式。
We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a $\C^*$-action. For varieties with an isolated singularity, we show that the presence of sufficiently many rational curves outside the singular point strongly affects the character of the singularity. This provides an explanation of classical results due to H.A. Schwartz and G.H. Halphen on polynomial solutions of the generalized Fermat equation.
A previous result of the authors with Chaput and Perrin states that the union of all rational curves of fixed degree passing through a Schubert variety in a homogeneous space G/P is again a Schubert variety. In this paper we identify this Schubert variety explicitly in terms of the Hecke product of Weyl group elements. We apply our result to give an explicit formula for any two-point Gromov-Witten invariant as well as a new proof of the quantum Chevalley formula and its equivariant generalization. We also recover a formula for the minimal degree of a rational curve between two given points in …
In this paper we prove the existence of rational homology balls smoothly embedded in regular neighborhoods of certain linear chains of smooth $2$-spheres by using techniques from minimal model program for 3-dimensional complex algebraic variety.
Let W be the germ of a smooth complex surface around an exceptional curve and let E be a rank 2 vector bundle on W. We study the cohomological properties of a finite sequence $E_i, 1 \leq i \leq t$ of rank 2 vector bundles canonically associated to E. We calculate numerical invariants of E in terms of the $E_i.$ If S is a compact complex smooth surface and E is a rank 2 bundle on the blow- up of S at a point, we show that all values of $c_2(E)-c_2(π_* E^{\vee\vee})$ that are numerically possible are actually attained.
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 non-singular, then this rational map is a morphism. As a consequence, we obtain that, if the surface is rational and we fix a divisor class containing a non-singular rational curve, then the scheme parametrizing rational curves on this class is irreducible. Further, if the class has non-negative self-intersection, then the scheme of rational curves has expecte…
Using ideas from the theory of tropical curves and degeneration, we prove that any Fano hypersurface (and more generally Fano complete intersections) is swept by at most quadratic rational curves.
We investigate the possible homological classes of rational curves on the moduli space $X_n=\bar{\mathcal{M}_{0,n}}$ of rational nodal curves with $n$ marked points. In the case of $X_5$ and $X_6$ the relevant homology classes belong to certain polyhedral cones which we describe explicitly. Not all homology classes in the cone correspond to rational curves, but all classes sufficiently far from the walls of the cone do. The proof for $X_5$ is explicit, the proof for $X_6$ was carried out using computers.
We give a characterization of the rational normal curve in terms of the rank function associated to a curve.
We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a generalization of the curve are also presented.
Approximating curves on real rational surfaces
本文给出实有理曲面上简单闭曲线可被光滑有理曲线逼近的拓扑充要条件，并研究了给定复自交数的逼近问题。
The stable rationality of components of the moduli space of (unparametrized) rational curves in projective $n$-space with fixed normal bundle is proved, provided these components dominate the moduli space of immersed rational curves in the plane.
We characterize plane rational curves of degree four with two or more inner Galois points. A computer verifies the existence of plane rational curves of degree four with three inner Galois points. This would be the first example of a curve with exactly three them. Our result implies that Miura's bound is sharp for rational curves.
Given a curve C on a projective nonsingular rational surface S, over an algebraically closed field of characteristic zero, we are interested in the set Omega_C of linear systems Lambda on S satisfying C is in Lambda, dim Lambda > 0, and the general member of Lambda is a rational curve. The main result of the paper gives a complete description of Omega_C and, in particular, characterizes the curves C for which Omega_C is non empty.
We prove that a semiring isomorphism between the rational function semifields of two tropical curves induces an expansive map between those tropical curves. This semiring isomorphism and the expansive map respect zeros and poles of rational functions with their degrees. As a corollary, we show that the automorphism group of a tropical curve is isomorphic to the $\boldsymbol{T}$-algebra automorphism group of its rational function semifield, where $\boldsymbol{T} := (\boldsymbol{R} \cup \{ -\infty \}, \operatorname{max}, +)$ is the tropical semifield. Finally, we describe all semiring automorphi…
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 with balanced normal bundle, and reprove some results on irreducibility of spaces of rational curves of low degree.
We provide a real analog of the Yau-Zaslow formula counting rational curves on $K3$ surfaces.
We proved the existence of rational curves in every linear system on a general K3 surface and that all rational curves in the hyperplane class are nodal on a general K3 surface of small genus.
In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential curves. Then we show that the implicitization problem of proper linear differential rational parametric equations can be solved by means of differential resultants. Furthermore, for linear differential curves, we give an algorithm to determine whether an implicitly given linear differential curve is unirational and, in the affirmative case, to compute a proper dif…
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 del Pezzo surface such that for every closed point $p\in X$, there is a smooth curve (locally analytically) passing through $p$, then $X$ contains at least one smooth rational curve.
The moduli spaces of trigonal curves of odd genus $g>4$ are proven to be rational.
We proved that every rational curves in the primitive class of a general K3 surface of any genus is nodal.
We prove that every curve on a rationally connected variety is algebraically equivalent to a (non-effective) integral sum of rational curves.
The moduli spaces of trigonal curves are proven to be rational when the genus is divisible by 4.
Nonlinear neighborhoods of smooth curves and arbitrarily negative Wahl defects
该研究建立亏格至少为2的光滑射影曲线的非线性邻域理论，将其开子集与圆盘乘积通过圆盘坐标非线性变换粘合，针对非有理二重点的正规复曲面奇点极小良预解式，给出Wahl亏格的有限公式，应用于超椭圆曲线与光滑平面曲线，证明Wahl主猜想不成立。
The aim of these notes is to give a introduction to the ideas and techniques of handling rational curves on varieties. The main emphasis is on varieties with many rational curves which are the higher dimensional analogs of rational curves and surfaces. The first six sections closely follow the lectures of J. Kollár given at the summer school ``Higher dimensional varieties and rational points''. The notes were written up by C. Araujo and substantially edited later. Revised version with 2 new sections.
We exploit an elementary specialization technique to study some properties of rational curves on index $n-1$ Fano $n$-folds. We prove a simple formula for counting rational curves passing through a suitable number of points in the case $n=3$. The arguments have immediate deformation theoretic consequences which translate to properties of the normal bundles of such curves.
Submission on request. This Master thesis from 2008 (University of Oslo, Norway) contains no new results, but it provides an overview of plane rational cuspidal curves, in particular curves of low degree. Please note that new results on this topic and the related topic of rational cuspidal curves on Hirzebruch surfaces have been published after this thesis was written.
The purpose of this article is to shed light on the question of how many and what kind of cusps a rational cuspidal curve on a Hirzebruch surface can have. We use birational transformations to construct rational cuspidal curves with four cusps on the Hirzebruch surfaces and find associated results for these curves.
We prove that the spaces of rational curves on del Pezzo surfaces are either irreducible or empty, with a unique exception.
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
This paper deals with rational curves and birational contractions on irreducible holomorphically symplectic manifold. We survey some recent results about minimal rational curves, their deformations, extremal rays associated with these curves, and the geometry of the Kähler cone.
In this series of three papers we start to investigate the rational Chow ring of the stack consisting of nodal curves of genus 0, in particular we determine completely the rational Chow ring of the substack consisting of curves with at most 3 nodes. In this first paper we construct the stack of rational nodal curves and its stratification by nodes and show that the map from the universal curve to the stack is not representable in the category of schemes.
According to a conjecture of H. Clemens, the dimension of the space of rational curves on a general projective hypersurface should equal the number predicted by a naïve dimension count. In the case of a general hypersurface of degree 7 in $\mathbb{P}^5$, the conjecture predicts that the only rational curves should be lines. This has been verified by Hana and Johnsen for rational curves of degree at most 15. Here we extend their results to show that no rational curves of degree 16 lie on a general heptic fourfold.
In this paper the notion of rational simple connectedness for the quintic Fano threefold $V_5\subset \mathbb{P}^6$ is studied and unirationality of the moduli spaces $\overline{M}_{0,0}^{\text{bir}}(V_5,d)$, with $d \ge 1$, is proved. Many further unirationality results for special moduli spaces of rational curves on quadric hypersurfaces and del Pezzo surfaces are obtained via explicit birational methods.
Let $(X,D)$ be a pair where $X$ is a projective variety. We study in detail how the behavior of rational curves on $X$ as well as the positivity of $-(K_X+D)$ and $D$ influence the behavior of rational curves on $D$. In particular we give criteria for uniruledness and rational connectedness of components of $D$.
This is a continuation of "Rational curves on hypersurfaces of low degree", math.AG/0203088. We prove that if d^2+d+1 < n and d > 2, then for a general hypersurface X_d in P^n of degree d, for each degree e the space of rational curves of degree e on X is itself a rationally connected variety.
Let $W$ be a subset of the set of real points of a real algebraic variety $X$. We investigate which functions $f: W \to \mathbb R$ are the restrictions of rational functions on $X$. We introduce two new notions: ${\it curve-rational \, functions}$ (i.e., continuous rational on algebraic curves) and ${\it arc-rational\, functions}$ (i.e., continuous rational on arcs of algebraic curves). We prove that under mild assumptions the following classes of functions coincide: continuous hereditarily rational (introduced recently by the first named author), curve-rational and arc-rational. In particular…
As in our previous work [1] we address the problem to determine the splitting of the normal bundle of rational curves. With apolarity theory we are able to characterize some particular subvarieties in some Hilbert scheme of rational curves, defined by the splitting type of the normal bundle and the restricted tangent bundle.
Normal bundles of rational curves in Grassmannians
本研究针对格拉斯曼流形中有理曲线法丛的平衡性质问题，证明了一般有理曲线的法丛可分解为次数相差至多2的线丛直和，补充了射影空间外该类性质的研究结论。
We determine all of lines in the moduli space $M$ of stable bundles for arbitrary rank and degree. A further application of minimal rational curves is also given in last section.
We give a simple proof of the statement that every rational curve in the primitive class of a general K3 surface is nodal.
We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve defined over Fq of geometric genus g and arithmetic genus $π$.
We construct the canonical structure of an irreducible projective variety on the set of connected curves of degree $d$ in $\Bbb P^n$ with rational components (some components can be multiple). The set of rational curves is open subset in the Zariski topology under this structure.
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