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An enriched count of nodal orbits in an invariant pencil of conics

Candace Bethea

TL;DR

The paper develops an equivariant, Burnside-ring-valued enrichment of the classical nodal-conic count in a general pencil of conics invariant under a finite group action on $\mathbb{CP}^2$. By associating to each nodal orbit a $G_t$-weight in $A(G_t)$ and inflating to $A(G)$, it proves a main theorem: the weighted sum of nodal orbits equals the base locus $[\Sigma]-\{*\}$ in $A(G)$ for all finite groups except $\mathbb{Z}/2\times\mathbb{Z}/2$, $A_4$, and $D_8$. The result generalizes the classical count (recovering 3 nodal conics when the action is trivial) and links to real-nodal signed counts, while also providing explicit counterexamples for the exceptional groups. The work thus advances equivariant enumerative geometry by yielding a Burnside-ring-valued invariant for nodal configurations in group-invariant pencils of plane conics.

Abstract

This work gives an equivariantly enriched count of nodal orbits in a general pencil of plane conics that is invariant under a linear action of a finite group on $\mathbb{CP}^2$. This is both inspired by and a departure from $R(G)$-valued enrichments such as Roberts's equivariant Milnor number and Damon's equivariant signature formula. Given a $G$-invariant general pencil of conics, the weighted sum of nodal orbits in the pencil is a formula in $A(G)$ in terms of the base locus considered as a $G$-set. We show this is true for all finite groups except $\mathbb{Z}/2\times \mathbb{Z}/2$, $A_4$, and $D_8$ and give counterexamples for the exceptional groups.

An enriched count of nodal orbits in an invariant pencil of conics

TL;DR

The paper develops an equivariant, Burnside-ring-valued enrichment of the classical nodal-conic count in a general pencil of conics invariant under a finite group action on . By associating to each nodal orbit a -weight in and inflating to , it proves a main theorem: the weighted sum of nodal orbits equals the base locus in for all finite groups except , , and . The result generalizes the classical count (recovering 3 nodal conics when the action is trivial) and links to real-nodal signed counts, while also providing explicit counterexamples for the exceptional groups. The work thus advances equivariant enumerative geometry by yielding a Burnside-ring-valued invariant for nodal configurations in group-invariant pencils of plane conics.

Abstract

This work gives an equivariantly enriched count of nodal orbits in a general pencil of plane conics that is invariant under a linear action of a finite group on . This is both inspired by and a departure from -valued enrichments such as Roberts's equivariant Milnor number and Damon's equivariant signature formula. Given a -invariant general pencil of conics, the weighted sum of nodal orbits in the pencil is a formula in in terms of the base locus considered as a -set. We show this is true for all finite groups except , , and and give counterexamples for the exceptional groups.
Paper Structure (5 sections, 6 theorems, 93 equations, 1 figure)

This paper contains 5 sections, 6 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Proof of the Classical Result
  4. An Equivariant Count of Orbits of Nodal Conics
  5. Counterexamples

Key Result

Theorem 1

Let $G$ be a finite group not isomorphic to $\mathbb{Z}/2\times\mathbb{Z}/2$, $A_4$, or $D_8$, and assume $G$ acts linearly on $\mathbb{C}\mathbb{P}^2$. Let $X$ be a $G$-invariant general pencil of conics in $\mathbb{C}\mathbb{P}^2$, and let $[\Sigma]$ in $A(G)$ represent the base locus of $X$. Then in $A(G)$. That is, there is a weighted count of orbits of nodal conics in $X$, valued in the Burns

Figures (1)

  • Figure 1: Distinct lines through $\Sigma=\{b_1, b_2, b_3, b_4\}$

Theorems & Definitions (15)

  • Theorem 1
  • Definition 3
  • Definition 4
  • Lemma 5
  • proof
  • Definition 6
  • Theorem 8
  • Lemma 10
  • proof
  • Lemma 11
  • ...and 5 more