Characteristic numbers of algebras

We introduce characteristic numbers of a finite commutative unital $\mathbb{C}$-algebra, which are numerical invariants arising from algebraic intersection theory. We characterize Gorenstein and local complete intersection algebras in terms of their characteristic numbers. We compute characteristic numbers for certain families of algebras. We show that characteristic numbers are constant on $\mathrm{Hilb}_d(\mathbb{A}^1)$, provide an explicit upper bound for characteristic numbers on the smoothable component of $\mathrm{Hilb}_d(\mathbb{A}^n)$ and an explicit lower bound for characteristic numbers on the Gorenstein locus of $\mathrm{Hilb}_d(\mathbb{A}^n)$ for $n \geq d-2$.