Relative node polynomials for plane curves
Relative node polynomials for plane curves
复制标题
平面曲线的相对节点多项式
DOI：
--
复制
发表时间：
2010
影响因子：
0.8
通讯作者：
Florian Block
中科院分区：
文献类型：
--
作者：
Florian Block
我们推广了S. Fomin和G. Mikhalkin关于多项式公式的Severi度。如果δ是固定的并且d足够大，则d次和δ节点的平面曲线的Severi簇的次数由d中的多项式给出。我们将这一结果推广到广义Severi品种parametrizing平面曲线，此外，满足切线条件的给定订单相对于一个给定的线。我们表明，这些品种的程度，适当地重新调整，给出了一个组合定义的“相对节点多项式”的切线订单，后者是足够大的。我们描述了一种计算任意δ的这些多项式的方法，并使用它来给出δ≤6的显式公式。我们还给出了多项式的阈值，并计算任何δ的前几个前导项。
We generalize the recent work of S. Fomin and G. Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and δ nodes is given by a polynomial in d, provided δ is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves that, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined “relative node polynomial” in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary δ, and use it to present explicit formulas for δ≤6. We also give a threshold for polynomiality, and compute the first few leading terms for any δ.