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Counting planar curves in $\mathbb{P}^3$ with degenerate singularities

Nilkantha Das, Ritwik Mukherjee

TL;DR

The paper derives explicit formulas for the characteristic numbers of planar degree $d$ curves in $\mathbb{P}^3$ with up to four codimension-sworth singularities, by counting curves whose image lies in a $\mathbb{P}^2$ and imposing nodes and more-degenerate singularities. The authors deploy a topological Euler-class method, extending Zinger’s framework and Basu–Mukherjee’s refinements, to account for degenerate boundary contributions across strata of equisingular loci. They provide recursive formulas for various singularity types (including $A_1$–type nodes and higher types $A_k$, $D_k$ with marked directions) and implement them computationally, with extensive verifications against Kleiman–Piene, Laarakker, Singh–Mukherjee, and Pandharipande’s BPS-number predictions. The results yield explicit polynomials in $d$ for many cases and align with known enumerations in low-degree scenarios, supporting conjectural aspects of BPS enumerativity in this geometric setting. Overall, the work extends planar-curve enumerative theory in $\mathbb{P}^3$ to include degenerations beyond nodes, enriching connections to moduli-space techniques and Gromov–Witten theory.

Abstract

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 $\mathbb{P}^3$.

Counting planar curves in $\mathbb{P}^3$ with degenerate singularities

TL;DR

The paper derives explicit formulas for the characteristic numbers of planar degree curves in with up to four codimension-sworth singularities, by counting curves whose image lies in a and imposing nodes and more-degenerate singularities. The authors deploy a topological Euler-class method, extending Zinger’s framework and Basu–Mukherjee’s refinements, to account for degenerate boundary contributions across strata of equisingular loci. They provide recursive formulas for various singularity types (including –type nodes and higher types , with marked directions) and implement them computationally, with extensive verifications against Kleiman–Piene, Laarakker, Singh–Mukherjee, and Pandharipande’s BPS-number predictions. The results yield explicit polynomials in for many cases and align with known enumerations in low-degree scenarios, supporting conjectural aspects of BPS enumerativity in this geometric setting. Overall, the work extends planar-curve enumerative theory in to include degenerations beyond nodes, enriching connections to moduli-space techniques and Gromov–Witten theory.

Abstract

In this paper, we consider the following question: how many degree curves are there in (passing through the right number of generic lines and points), whose image lies inside a , having nodes and one singularity of codimension . We obtain an explicit formula for this number when (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 with a cusp. We also verify some of the numbers predicted by the conjecture made by Pandharipande, regarding the enumerativity of BPS numbers for .

Paper Structure

This paper contains 20 sections, 9 theorems, 276 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setup and Notation
  3. Cohomology ring structure of projective fiber bundles
  4. Intersection Numbers
  5. Recursive Formulas
  6. Proof of the recursive formulas
  7. Proof of \ref{['na1_again']} and \ref{['na1_delta']}: computation of $N(A_1^{\delta}A_1)$ when $0\leq \delta \leq 3$
  8. Local degree of a smooth map
  9. Proof of \ref{['npa1']}: computation of $N(A_1^{\delta} \mathcal{P} A_1)$ when $0\leq \delta \leq 2$
  10. Proof of \ref{['npa2']}: computation of $N(A_1^{\delta} \mathcal{P} A_2)$ when $0\leq \delta \leq 2$
  11. Proof of \ref{['npa3']}: computation of $N(A_1^{\delta} \mathcal{P} A_3)$ when $0\leq \delta \leq 1$
  12. Proof of \ref{['npa4']}: computation of $N(\mathcal{P} A_4)$
  13. Proof of \ref{['npd4']}: computation of $N(\mathcal{P} D_4)$
  14. Verification with other results and low degree checks
  15. Verification with S. Kleiman and R. Piene's result
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $\mathfrak{X}$ be a singularity of codimension $c_{\mathfrak{X}}$ and $\delta$ a non negative integer. We obtain an explicit formula for $N_d^{\textnormal{Planar}, \mathbb{P}^3}(A_1^{\delta} \mathfrak{X}, r,s)$, when $\delta + c_{\mathfrak{X}} \leq 4$, provided $d \geq d_{\textnormal{min}}$, whe

Figures (3)

  • Figure 1: Two nodes colliding into a tacnode
  • Figure 2: Three nodes colliding into an $A_5$-singularity
  • Figure 3: Three nodes colliding into a triple point

Theorems & Definitions (31)

  • Definition 1.1
  • Remark 1
  • Theorem 1.2
  • Remark 2
  • Theorem 5.1
  • Remark 3
  • Theorem 5.2
  • Theorem 5.3
  • Remark 4
  • Theorem 5.4
  • ...and 21 more