论文标题
$ \ mathbb {p}^2 $的移动家族中的有理尖曲线
Rational Cuspidal Curves in a moving family of $\mathbb{P}^2$
论文作者
论文摘要
在本文中，我们获得了$ \ mathbb {p}^3 $具有CUSP的理性学位d曲线数量的公式，其图像位于$ \ mathbb {p}^2 $中，并通过$ r $ $ lines和$ s $ lines和$ s $点（其中$ r + 2s = 3d + 1 $）。这个问题可以看作是在$ \ mathbb {p}^2 $中计算有理尖曲线的经典问题的家庭版本，该曲线已由Z. Ran，R。Pandharipande和A. Zinger进行了研究。我们通过计算相关捆绑包的Euler类，然后找出对Euler类的相应退化贡献来获得此数字。我们使用的方法紧密基于A. Zinger和I. Biswas，S。D'Mello，R。Mukherjee和V. Pingali。我们还验证了我们对理性的尖齿平面立方体和四重奏的特征数量的答案与N. Das和第一作者获得的答案是一致的，在该答案中，他们在$ \ mthbb {p}^3 $中以$ \ mathbb {p}^$ for One cusp（for One cusp（for $δ）（for $δ\ leauq 2 $ leq leq 2 $）。
In this paper we obtain a formula for the number of rational degree d curves in $\mathbb{P}^3$ having a cusp, whose image lies in a $\mathbb{P}^2$ and that passes through $r$ lines and $s$ points (where $r + 2s = 3d + 1$). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in $\mathbb{P}^2$, which has been studied earlier by Z. Ran, R. Pandharipande and A. Zinger. We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger and I. Biswas, S. D'Mello, R. Mukherjee and V. Pingali. We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author, where they compute the characteristic number of $δ$-nodal planar curves in $\mathbb{P}^3$ with one cusp (for $δ\leq 2$).
