Betti numbers of full Perazzo algebras

TL;DR

This work identifies a structural mechanism for full Perazzo algebras: each such artinian Gorenstein algebra $A_F$ is the doubling of a $0$-dimensional scheme $Z_F$ consisting of $n+1$ double points, and it derives explicit graded Betti numbers for a minimal free resolution of $A_F$ by combining the resolution of $R/I(Z_F)$ with the doubling construction via a mapping cone. The key method hinges on expressing the Macaulay dual generator in a form compatible with doubling and then transferring Betti information through homological constructions, yielding a self-dual Betti table whose nonzero entries in the first nonzero row have closed-form binomial expressions in terms of $d$, $n$, and $m$. These results illuminate Lefschetz-type properties of Perazzo algebras and pave the way for characterizing which Perazzo algebras arise as doublings of zero-dimensional schemes. The findings provide concrete, computable invariants and suggest natural generalizations to broader classes of Perazzo algebras.

Abstract

In this paper we prove that any full Perazzo algebra $A_F$, whose Macaulay dual generator is a Perazzo form $F\in K[X_0,\dots,X_n,U_1,\dots,U_m]_d$ with $n+1 = \binom{d+m-2}{m-1}$, is the doubling of a 0-dimensional scheme in $\PP^{n+m}$ and we compute the graded Betti numbers of a minimal free resolution of $A_F$.

Theorems & Definitions (31)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Example 2.4
• Definition 2.5
• Definition 2.6
• Example 2.7
• Lemma 3.1
• proof
• Proposition 3.3