The moduli space of commutative algebras of finite rank

Abstract

The moduli space of rank-$n$ commutative algebras equipped with an ordered basis is an affine scheme $\frakB_n$ of finite type over $\Z$, with geometrically connected fibers. It is smooth if and only if $n \le 3$. It is reducible if $n \ge 8$ (and the converse holds, at least if we remove the fibers above $2$ and $3$). The relative dimension of $\frakB_n$ is $\frac{2}{27} n^3 + O(n^{8/3})$. The subscheme parameterizing \'etale algebras is isomorphic to $\GL_n/S_n$, which is of dimension only $n^2$. For $n \ge 8$, there exist algebras that are not limits of \'etale algebras. The dimension calculations lead also to new asymptotic formulas for the number of commutative rings of order $p^n$ and the dimension of the Hilbert scheme of $n$ points in $d$-space for $d \ge n/2$.