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Enumeration of rational plane curves tangent to a smooth cubic. (English) Zbl 1157.14040

Let \(D\) be a complex plane projective (reduced) curve. Given two tuples of integers \(\alpha=(\alpha_1,\alpha_2,\dots)\), \(\beta=(\beta_1,\beta_2,\dots)\) an old question of enumerative geometry is:
How many plane rational curves of degree \(d\) meet \(D\) at \(\alpha_k\) fixed points with contact order \(k\) and \(\beta_k\) non-fixed points with contact order \(k\)?
(All contacts with \(D\) are assumed at the unibranch points of \(D\). To make the number of rational curves finite one imposes the condition: the curves pass through \(3d-1-\sum k\alpha_k-\sum(k-1)\beta_k\) general points).
The problem is very difficult and only some particular cases are known. The case \(\alpha=0\), \(\beta=(3d,0,\dots)\) was solved by Kontsevich, the case \(D=\) line by Caporaso-Harris, some cases with \(D=\) conic were treated by Vakil and Gathmann.
The authors treat the case \(D=\) cubic and \(\alpha,\beta\) arbitrary, except for \((\alpha,\beta)=(0,e_{3d})\).
The method uses several ingredients. A WDVV equation is used to provide relations among the invariants. This brings some non-enumerative contributions. The authors relate these contributions to some enumerative invariants. Further, the Caporaso-Harris recursion is used.
An interesting result is that the computation of all such numbers \(N_d(\alpha,\beta)\) is reduced to a smaller set of numbers \(M_d(\alpha+\beta)\) by \[ N_d(\alpha,\beta)=\prod k^{\beta_k}(\sum k\beta_k)\frac{(\sum\beta_k-1)!}{\prod(\beta_k!)}M_d(\alpha+\beta) \] and the later numbers \(M_d(\alpha+\beta)\) can be defined independently of the initial ones.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H45 Special algebraic curves and curves of low genus
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References:

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