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

Abstract

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