Quadratic linear strands of prime ideals

TL;DR

This paper establishes optimal bounds on the quadratic generators of prime ideals in polynomial rings, showing they are limited by the ideal's height and providing examples achieving these bounds.

Contribution

It provides sharp, height-dependent bounds on the quadratic strand of prime ideals' resolutions, including existence of ideals attaining these bounds.

Findings

Prime ideals of height h have at most h^2 quadratic minimal generators.

Existence of prime ideals minimally generated by h^2 quadrics.

Bounds are optimal and depend only on the height of the prime ideal.

Abstract

We prove sharp estimates on the quadratic strand of the resolution of any homogeneous prime ideal in a standard graded polynomial ring over an arbitrary field. Our bounds only depend on the height of the prime ideal, and they are optimal since for every $h \geq 1$ we show that there exists a prime ideal of height $h$ achieving them. In particular, we show that a prime ideal of height $h$ can contain at most $h^2$ quadratic minimal generators, and that there exists a prime ideal minimally generated by $h^2$ quadrics.