The minimal number of generators of a Togliatti system

TL;DR

This paper determines bounds on the number of generators for minimal smooth monomial Togliatti systems, classifies systems near these bounds, and explores stability properties of associated bundles.

Contribution

It provides the first comprehensive bounds and classifications for minimal smooth monomial Togliatti systems across all degrees and variables, including stability analysis.

Findings

Bounds on the number of generators are established for all degrees and variables.

Classification of systems reaching or near the bounds is achieved.

Gaps in possible numbers of generators are identified for higher dimensions.

Abstract

We compute the minimal and the maximal bound on the number of generators of a minimal smooth monomial Togliatti system of forms of degree $d$ in $n+1$ variables, for any $d\ge 2$ and $n\geq 2$. We classify the Togliatti systems with number of generators reaching the lower bound or close to the lower bound. We then prove that if $n=2$ (resp $n=2,3$) all range between the lower and upper bound is covered, while if $n\geq 3$ (resp. $n\ge 4$) there are gaps if we only consider smooth minimal Togliatti systems (resp. if we avoid the smoothness hypothesis). We finally analyze for $n=2$ the Mumford-Takemoto stability of the syzygy bundle associated to smooth monomial Togliatti systems.