平面曲线的标号楼层图
Labeled floor diagrams for plane curves
摘要：
楼层图是由E. Brugalle和第二作者。热带几何参数导致组合描述（普通和相对）Gromov-Witten不变量的射影空间的地板图和他们的推广。在许多情况下，这些描述可以用于获得相应枚举不变量的显式（直接或递归）公式。特别是，我们使用这种方法来枚举有理曲线的给定程度通过一个集合的点在复平面上，并具有最大的切线到一个给定的线。组合方法的另一个应用是P. Di Francesco-C. Itzykson和L. Goettsche指出，在一个固定的余属的情况下，通过适当的许多泛点的d次平面曲线的数量由d中的多项式给出，假设d足够大。此外，证明提供了一种计算这些“节点多项式”的方法。“标记楼层图是通过用整数1，.，d以与取向相容的方式。我们发现，标记的地板图属0是equinumerous标记树，因此计算由著名的凯莱公式。相应的双射导致解释的Kontsevich数（属0格罗莫夫-威滕不变量的射影平面）在某些统计方面的树木。
英文摘要：
Floor diagrams are a class of weighted oriented graphs introduced by E. Brugalle and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative) Gromov-Witten invariants of projective spaces in terms of floor diagrams and their generalizations. In a number of cases, these descriptions can be used to obtain explicit (direct or recursive) formulas for the corresponding enumerative invariants. In particular, we use this approach to enumerate rational curves of given degree passing through a collection of points on the complex plane and having maximal tangency to a given line. Another application of the combinatorial approach is a proof of a conjecture by P. Di Francesco -- C. Itzykson and L. Goettsche that in the case of a fixed cogenus, the number of plane curves of degree d passing through suitably many generic points is given by a polynomial in d, assuming that d is sufficiently large. Furthermore, the proof provides a method for computing these "node polynomials." A labeled floor diagram is obtained by labeling the vertices of a floor diagram by the integers 1,...,d in a manner compatible with the orientation. We show that labeled floor diagrams of genus 0 are equinumerous to labeled trees, and therefore counted by the celebrated Cayley's formula. The corresponding bijections lead to interpretations of the Kontsevich numbers (the genus-0 Gromov-Witten invariants of the projective plane) in terms of certain statistics on trees.
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