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Counting Curves with Tangencies

Indranil Biswas, Apratim Choudhury, Ritwik Mukherjee, Anantadulal Paul

TL;DR

This work introduces a novel viewpoint that treats tangency to a divisor as a limit of transverse intersections, enabling explicit, non-recursive polynomial-in-$d$ counts for smooth, nodal, and cuspidal plane curves with prescribed tangencies. The authors develop a collision-lemma–driven intersection theory framework to derive precise formulas for counting smooth curves with multiple tangencies, as well as $1$-nodal and $1$-cuspidal curves, including degenerations such as tacnodes. They extend the method to stable maps, producing concrete computations for rational curves with first-order tangency, and provide extensive low-degree checks against Caporaso–Harris, Ernström–Kennedy, and Gathmann. In addition, the paper offers multiple complementary approaches to count rational curves with tangency, including moving domain vs moving target points and WDVV-type relations, and connects with AiM_m_fold_pt to handle higher-fold singularities. Overall, the results yield a versatile toolkit for enumerating tangent incidences in plane curves and stable maps, with explicit formulas and extensive verifications across several singularity types and degrees.

Abstract

Interpreting tangency as a limit of two transverse intersections, we obtain a concrete formula to enumerate smooth degree $d$ plane curves tangent to a given line at multiple points with arbitrary order of tangency. Extending that idea, we then enumerate curves with one node with multiple tangencies to a given line of any order. Subsequently, we enumerate curves with one cusp, that are tangent to first order to a given line at multiple points. We also present a new way to enumerate curves with one node; it is interpreted as a degeneration of a curve tangent to a given line. That method is extended to enumerate curves with two nodes, and also curves with one tacnode are enumerated. In the final part of the paper, it is shown how this idea can be applied in the setting of stable maps and perform a concrete computation to enumerate rational curves with first-order tangency. A large number of low degree cases have been worked out explicitly.

Counting Curves with Tangencies

TL;DR

This work introduces a novel viewpoint that treats tangency to a divisor as a limit of transverse intersections, enabling explicit, non-recursive polynomial-in- counts for smooth, nodal, and cuspidal plane curves with prescribed tangencies. The authors develop a collision-lemma–driven intersection theory framework to derive precise formulas for counting smooth curves with multiple tangencies, as well as -nodal and -cuspidal curves, including degenerations such as tacnodes. They extend the method to stable maps, producing concrete computations for rational curves with first-order tangency, and provide extensive low-degree checks against Caporaso–Harris, Ernström–Kennedy, and Gathmann. In addition, the paper offers multiple complementary approaches to count rational curves with tangency, including moving domain vs moving target points and WDVV-type relations, and connects with AiM_m_fold_pt to handle higher-fold singularities. Overall, the results yield a versatile toolkit for enumerating tangent incidences in plane curves and stable maps, with explicit formulas and extensive verifications across several singularity types and degrees.

Abstract

Interpreting tangency as a limit of two transverse intersections, we obtain a concrete formula to enumerate smooth degree plane curves tangent to a given line at multiple points with arbitrary order of tangency. Extending that idea, we then enumerate curves with one node with multiple tangencies to a given line of any order. Subsequently, we enumerate curves with one cusp, that are tangent to first order to a given line at multiple points. We also present a new way to enumerate curves with one node; it is interpreted as a degeneration of a curve tangent to a given line. That method is extended to enumerate curves with two nodes, and also curves with one tacnode are enumerated. In the final part of the paper, it is shown how this idea can be applied in the setting of stable maps and perform a concrete computation to enumerate rational curves with first-order tangency. A large number of low degree cases have been worked out explicitly.
Paper Structure (21 sections, 20 theorems, 101 equations, 8 figures)

This paper contains 21 sections, 20 theorems, 101 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Comparison with other methods
  4. A few remarks on intersection theory
  5. The collision lemma
  6. Counting smooth curves with multiple tangencies
  7. Counting $\boldsymbol{ 1}$-nodal curves with multiple tangencies
  8. Counting $\boldsymbol{ 1}$-cuspidal curves with multiple tangencies of first order
  9. Counting singular curves
  10. Counting $\boldsymbol{ 1}$-nodal curves
  11. Counting $\boldsymbol{ 2}$-nodal curves
  12. Counting $\boldsymbol{ 1}$-tacnodal curves
  13. Counting rational curves
  14. A review of Gathmann's approach to count curves with tangencies
  15. A new method to count rational curves with first-order tangency
  16. ...and 6 more sections

Key Result

Lemma 5.1

The cohomology class $(a_1+a_2-y_1)$ restricted to $X$ is equal to the Poincaré dual of $Y$ in $X$.

Figures (8)

  • Figure 1: Tangency as a limit of transverse intersection.
  • Figure 2: Two nodes on the line limiting to a tacnode.
  • Figure 3: Stable maps intersecting a line at two points.
  • Figure :
  • Figure :
  • ...and 3 more figures

Theorems & Definitions (28)

  • Lemma 5.1: collision lemma
  • Remark 5.2
  • Lemma 6.1
  • Theorem 6.2
  • Proposition 6.3
  • Remark 6.4
  • Theorem 6.5
  • Remark 6.6
  • Definition 7.1
  • Proposition 7.2
  • ...and 18 more