Computing mathbb {A}^1-Euler numbers with Macaulay2

Abstract

We use Macaulay2 for several enriched counts in {text {GW}}(k). First, we compute the count of lines on a general cubic surface using Macaulay2 over mathbb {F}_p in {text {GW}}(mathbb {F}_p) for p a prime number and over mathbb {Q} in {text {GW}}(mathbb {Q}). This gives a new proof for the fact that the mathbb {A}^1-Euler number of {text {Sym}}^3mathcal {S}^*rightarrow {text {Gr}}(2,4) is 15langle 1rangle +12langle -1rangle . Then, we compute the count of lines in mathbb {P}^3 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 mathbb {A}^1-Milnor numbers.