Betti numbers of Bresinsky's curves in $\mathbb{A}^{4}$

TL;DR

This paper investigates the Betti numbers of Bresinsky's monomial curves in four-dimensional affine space, demonstrating their unbounded growth and providing explicit minimal free resolutions.

Contribution

It extends previous results by showing all Betti numbers are unbounded for Bresinsky's curves and constructs explicit minimal free resolutions.

Findings

All Betti numbers are unbounded for Bresinsky's curves.

An explicit minimal free resolution is constructed.

The unboundedness property applies to the entire Betti sequence.

Abstract

Bresinsky defined a class of monomial curves in $\mathbb{A}^{4}$ with the property that the minimal number of generators or the first Betti number of the defining ideal is unbounded above. We prove that the same behaviour of unboundedness is true for all the Betti numbers and construct an explicit minimal free resolution for this class.