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Computing A1-Euler numbers with Macaulay2

Sabrina Pauli

TL;DR

The paper develops and applies Macaulay2-based computations to enrich classical enumerative geometry with A1- (Grothendieck–Witt) data. By computing the A1-Euler numbers for vector bundles on Grassmannians and related bundles, it derives enriched counts for lines on cubic surfaces, lines meeting general lines, and singular elements in pencils, producing explicit GW-classes such as e^{A1}(Sym^3 S*) = 15⟨1⟩+12⟨-1⟩ and related hyperbolic multiples. It also formalizes and computes EKL-forms and A1-Milnor numbers for singularities, including Du Val types, via dedicated EKL-code and discusses the connections to existing theories of Witt-valued characteristic classes. The work provides concrete code and examples that demonstrate how Macaulay2 can operationalize A1-enrichment in enumerative problems, offering new proofs and quantitative invariants that refine complex and real counts in GW(k).

Abstract

We use Macaulay2 for several enriched counts in GW(k). First, we compute the count of lines on a general cubic surface using Macaulay2 over Fp in GW(Fp) for p a prime number and over the rational numbers Q in GW(Q). This gives a new proof for the fact that the count of lines on a cubic surface is 3+12h in GW(k) where h denotes the hyperbolic form. Then, we compute the count of lines in P3 meeting 4 general lines, the count of lines on a quadratic surface meeting one general line and the count of singular elements in a pencil of degree d-surfaces. Finally, we provide code to compute the EKL-form and compute several A1-Milnor numbers.

Computing A1-Euler numbers with Macaulay2

TL;DR

The paper develops and applies Macaulay2-based computations to enrich classical enumerative geometry with A1- (Grothendieck–Witt) data. By computing the A1-Euler numbers for vector bundles on Grassmannians and related bundles, it derives enriched counts for lines on cubic surfaces, lines meeting general lines, and singular elements in pencils, producing explicit GW-classes such as e^{A1}(Sym^3 S*) = 15⟨1⟩+12⟨-1⟩ and related hyperbolic multiples. It also formalizes and computes EKL-forms and A1-Milnor numbers for singularities, including Du Val types, via dedicated EKL-code and discusses the connections to existing theories of Witt-valued characteristic classes. The work provides concrete code and examples that demonstrate how Macaulay2 can operationalize A1-enrichment in enumerative problems, offering new proofs and quantitative invariants that refine complex and real counts in GW(k).

Abstract

We use Macaulay2 for several enriched counts in GW(k). First, we compute the count of lines on a general cubic surface using Macaulay2 over Fp in GW(Fp) for p a prime number and over the rational numbers Q in GW(Q). This gives a new proof for the fact that the count of lines on a cubic surface is 3+12h in GW(k) where h denotes the hyperbolic form. Then, we compute the count of lines in P3 meeting 4 general lines, the count of lines on a quadratic surface meeting one general line and the count of singular elements in a pencil of degree d-surfaces. Finally, we provide code to compute the EKL-form and compute several A1-Milnor numbers.

Paper Structure

This paper contains 15 sections, 1 theorem, 17 equations, 2 tables.

Table of Contents

  1. Introduction
  2. $\mathbb{A}^1$-Euler numbers
  3. Definition of the $\mathbb{A}^1$-Euler number
  4. Cubic Surfaces
  5. The trace form
  6. Lines meeting four general lines in $\mathbb{P}^3$
  7. Lines on a degree 2 hypersurface in $\mathbb{P}^3$ meeting 1 general line
  8. Signature of $e^{\mathbb{A}^1}(\mathcal{E})=e^{\mathbb{A}^1}(\operatorname{Sym}^3\mathcal{S}^*_{\mathbb{Q}})$
  9. Singular elements on a pencil of degree $d$ hypersurfaces in $\mathbb{P}^3$
  10. EKL-class
  11. EKL-code
  12. $\mathbb{A}^1$-Milnor numbers
  13. Du Val Singularities
  14. An example of lines on a cubic
  15. More $\mathbb{A}^1$-Milnor numbers

Key Result

Lemma 2.9

The coordinates defined by the basis eq: basis tangent (which are equal to the coordinates chosen in subsection subsection: cubic surfaces) and the trivialization of $\mathcal{E}_3\vert_U$ defined by the basis eq: basis bundle are compatible with the canonical relative orientation of $\mathcal{E}_3$

Theorems & Definitions (20)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Lemma 2.9
  • proof
  • ...and 10 more