论文标题
在$ \ mathbb {p}^3 $中计数平面曲线，并带有堕落的奇异性
Counting planar curves in $\mathbb{P}^3$ with degenerate singularities
论文作者
论文摘要
在本文中，我们考虑以下问题：$ \ mathbb {p}^3 $中有多少个度$ d $曲线（通过正确数量的通用线和点），其图像在$ \ mathbb {p}^2 $内，具有$Δ$ nodes，并且具有$δ$ nodes和一个奇特的conimension $ k $。当$δ+k \ leq 4 $时，我们获得了该数字的明确公式（即，奇异性的总编成不超过四个）。我们使用拓扑方法来计算对欧拉类的退化贡献。它是A. Zinger的论文中的方法的扩展，并由S. Basu和第二作者进一步追求。使用这种方法，当存在的奇异性比节点（例如cusps，tacnodes和Triple点）更简单时，我们获得了公式。当奇异性仅是节点时，我们已经证实了我们的答案与S. Kleiman和R. Piene和T. Laarakker获得的答案是一致的。我们还验证了我们对具有尖尖的平面立方体数量的答案，以及带有两个节点的平面四分之一的数量，一个尖齿与R. Singh和第二作者获得的答案是一致的，他们在其中计算了$ \ sathbb {p}^3 $ cusp的$ \ mathbb {p}^3 $的理性平面曲线的特征数量。我们还验证了Pandharipande提出的一些数字，内容涉及$ \ Mathbb {p}^3 $的BPS数字的枚举。
In this paper, we consider the following question: how many degree $d$ curves are there in $\mathbb{P}^3$ (passing through the right number of generic lines and points), whose image lies inside a $\mathbb{P}^2$, having $δ$ nodes and one singularity of codimension $k$. We obtain an explicit formula for this number when $δ+k \leq 4$ (i.e. the total codimension of the singularities is not more than four). We use a topological method to compute the degenerate contribution to the Euler class; it is an extension of the method that originates in a paper by A. Zinger and which is further pursued by S. Basu and the second author. Using this method, we have obtained formulas when the singularities present are more degenerate than nodes (such as cusps, tacnodes and triple points). When the singularities are only nodes, we have verified that our answers are consistent with those obtained by by S. Kleiman and R. Piene and by T. Laarakker. We also verify that our answer for the characteristic number of planar cubics with a cusp and the number of planar quartics with two nodes and one cusp is consistent with the answer obtained by R. Singh and the second author, where they compute the characteristic number of rational planar curves in $\mathbb{P}^3$ with a cusp. We also verify some of the numbers predicted by the conjecture made by Pandharipande, regarding the enumerativity of BPS numbers for $\mathbb{P}^3$.
