On the projective dimension of $5$ quadric almost complete intersections with low multiplicities

TL;DR

This paper characterizes the degree two components of certain prime ideals and establishes a tight bound on the projective dimension of five-quadratic almost complete intersections with low multiplicity.

Contribution

It provides a finite classification of degree two parts of ideals primary to a specific prime and derives a new bound on the projective dimension for a class of almost complete intersections.

Findings

Finite characterization of degree two components for ideals with multiplicity ≤ 3.

Tight bound on projective dimension for five-quadratic almost complete intersections.

Applicable over algebraically closed fields.

Abstract

Let $S$ be a polynomial ring over an algebraic closed field $k$ and $ \mathfrak p =(x,y,z,w) $ a homogeneous height four prime ideal. We give a finite characterization of the degree two component of ideals primary to $\mathfrak p$, with multiplicity $e \leq 3$. We use this result to give a tight bound on the projective dimension of almost complete intersections generated by five quadrics with $e \leq 3$.