# The Eisenbud-Green-Harris Conjecture for Defect Two Quadratic Ideals

## TL;DR

This paper proves the Eisenbud-Green-Harris conjecture for a specific case involving five quadrics in a polynomial ring, confirming the conjecture's validity in this scenario.

## Contribution

It provides a complete proof of the EGH conjecture for ideals generated by seven quadrics in five variables with equal degrees, a previously unresolved case.

## Key findings

- EGH conjecture holds for n=5 with all generators quadratic
- Complete proof for ideals generated by n+2 quadrics containing a regular sequence
- Confirms EGH conjecture in this specific quadratic case

## Abstract

The Eisenbud-Green-Harris (EGH) conjecture states that a homogeneous ideal in a polynomial ring $K[x_1,\,\ldots,\,x_n]$ over a field $K$ that contains a regular sequence $f_1,\,\ldots,\, f_n$ with degrees $a_i$, $i=1,\,\ldots,\,n$ has the same Hilbert function as a lex-plus-powers ideal containing the powers $x_i^{a_i}$, $i=1,\,\ldots,\,n$. In this paper, we discuss a case of the EGH conjecture for homogeneous ideals generated by $n+2$ quadrics containing a regular sequence $f_1,\, \ldots, \, f_n$ and give a complete proof for EGH when $n=5$ and $a_1=\cdots=a_5=2$.