The enumerative geometry of rational and elliptic curves in projective space
The enumerative geometry of rational and elliptic curves in projective space
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射影空间有理曲线和椭圆曲线的枚举几何
DOI：
10.1515/crll.2000.094
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
R. Vakil
中科院分区：
文献类型：
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作者：
R. Vakil
研究了P^n中与固定的一般线性空间相交并与固定的超平面H相切的参数化d次有理曲线和椭圆曲线的几何性质，其中超平面H具有固定的重数且沿H的固定的一般线性子空间沿着.作为应用，我们推导出递归公式的数量时，这样的曲线的数量是有限的。这些递归公式只需要一个输入作为“种子数据”：P^1中有一条线通过两个点。这些数可以被看作是P^n中d次有理或椭圆曲线的希尔伯特方案上的各种圈的顶交积，或者是$\mbar_0（P^n，d）$或$\mbar_1（P^n，d）$的某些分量上的顶交积，因此给出了这些对象的Chow环（以及拓扑）的信息。该公式也可以解释为在适当的希尔伯特方案或稳定映射空间的Chow环（不一定在顶层）中的等式。特别地，给出了P^n中与各种固定的一般线性空间相交的有理曲线和椭圆曲线的计数算法。(The亏格0的数是由Kontsevich-Manin较早发现的，亏格1的数是由Ran和Caporaso-Harris发现的，n = 2时是由Getzler独立发现的。
We study the geometry of varieties parametrizing degree d rational and elliptic curves in P^n intersecting fixed general linear spaces and tangent to a fixed hyperplane H with fixed multiplicities along fixed general linear subspaces of H. As an application, we derive recursive formulas for the number of such curves when the number is finite. These recursive formulas require as ``seed data'' only one input: there is one line in P^1 through two points. These numbers can be seen as top intersection products of various cycles on the Hilbert scheme of degree d rational or elliptic curves in P^n, or on certain components of $\mbar_0(P^n,d)$ or $\mbar_1(P^n,d)$, and as such give information about the Chow ring (and hence the topology) of these objects. The formula can also be interpreted as an equality in the Chow ring (not necessarily at the top level) of the appropriate Hilbert scheme or space of stable maps. In particular, this gives an algorithm for counting rational and elliptic curves in P^n intersecting various fixed general linear spaces. (The genus 0 numbers were found earlier by Kontsevich-Manin, and the genus 1 numbers were found for n=2 by Ran and Caporaso-Harris, and independently by Getzler for n=3.)