Computing A1-Euler numbers with Macaulay2

TL;DR

This paper demonstrates the use of Macaulay2 to compute enriched enumerative invariants like A1-Euler numbers and Milnor numbers, providing new proofs and computational tools in algebraic geometry.

Contribution

It introduces methods to compute A1-Euler numbers and Milnor numbers using Macaulay2, including new proofs for classical enumerative counts.

Findings

Computed lines on cubic surfaces over Fp and Q in GW(k).

Provided code for computing EKL-forms and A1-Milnor numbers.

Validated counts of lines meeting conditions in projective space.

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.