The number of plane conics that are five-fold tangent to a given curve
The number of plane conics that are five-fold tangent to a given curve
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与给定曲线五重相切的平面二次曲线的数量
作者：
A. Gathmann
给定d次平面曲线Y，我们计算与Y相切5倍的不可约平面二次曲线的个数和。Vainsencher以前用经典方法研究过这个问题，但由于计算产生了太多无法分析的非枚举校正项而无法解决。在我们目前的方法中，我们用可以直接计算的相对Gromov-Witten不变量来表示数和。作为应用，我们把给定的K3曲面看作$\mathbb{P}^2$沿一条六分形曲线分支的双覆盖。我们计算了$\mathbb{P}^2$中二次曲线的回拉同调类中K3曲面上的有理曲线的个数，并与相应的yu - zaslow K3不变量进行了比较。这给出了一个非基元同调类的K3不变量的例子。
Given a general plane curve Y of degree d, we compute the number nd of irreducible plane conics that are five-fold tangent to Y. This problem has been studied before by Vainsencher using classical methods, but it could not be solved because the calculations produced too many non-enumerative correction terms that could not be analyzed. In our current approach, we express the number nd in terms of relative Gromov–Witten invariants that can then be directly computed. As an application, we consider the K3 surface given as the double cover of $\mathbb{P}^2$ branched along a sextic curve. We compute the number of rational curves in this K3 surface in the homology class that is the pull-back of conics in $\mathbb{P}^2$, and compare this number with the corresponding Yau–Zaslow K3 invariant. This gives an example of such a K3 invariant for a non-primitive homology class.