We obtain an explicit formula for the characteristic number of degree curves in with prescribed singularities (of type ) that are tangent to a given line. The formula is in terms of the characteristic number of curves with exactly those singularities. We are not aware of any explicit formula to enumerate plane curves of degree with any number of singularities (beyond codimension 8); however, combined with the results of S. Basu and R. Mukherjee (, , and ), this gives us a complete formula for the characteristic number of curves with -nodes and one singularity of type , tangent to a given line, provided . We use a topological method to compute the degenerate contribution to the Euler class. We have made several low degree checks to verify special cases of our result. When the singularities are only nodes, we have verified that our numbers are logically consistent with those computed by L. Caporaso and J. Harris (). We also verify that our answers for the characteristic number of cubics with a cusp tangent to a given line and the characteristic number of quartics with two nodes and a cusp, tangent to a given line is logically consistent with the characteristic number of rational cubics and quartics tangent to a given line that was computed by L. Ernström and G. Kennedy ().

中文翻译：

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计算具有规定切线的奇异曲线

我们获得了一个明确的公式，用于表示与给定线相切的指定奇点（类型为 ）的度曲线的特征数。该公式是根据具有这些奇点的曲线的特征数来计算的。我们不知道有任何明确的公式可以枚举具有任意数量奇点（余维 8 之外）的平面曲线；然而，结合 S. Basu 和 R. Mukherjee (, , 和 ) 的结果，这为我们提供了一个完整的公式，用于计算具有 - 节点和一个与给定线相切的类型奇点的曲线的特征数。我们使用拓扑方法来计算对欧拉类的简并贡献。我们进行了几次低度检查来验证我们结果的特殊情况。当奇点只是节点时，我们已经验证我们的数字在逻辑上与 L. Caporaso 和 J. Harris () 计算的数字一致。我们还验证了我们对具有与给定线相切的尖点的三次方特征数和具有两个节点和一个尖点与给定线相切的四次方特征数的答案在逻辑上与有理三次方和四次方的特征数一致与由 L. Ernström 和 G. Kennedy () 计算的给定线相切。