A Survey on The Eisenbud-Green-Harris Conjecture

TL;DR

This survey compiles and discusses various results related to the Eisenbud-Green-Harris conjecture, a generalization of Macaulay's theorem concerning Hilbert functions of homogeneous ideals.

Contribution

It provides a comprehensive overview of known results and approaches regarding the Eisenbud-Green-Harris conjecture, highlighting progress and open problems.

Findings

Multiple results supporting the EGH conjecture in special cases

Different proof techniques and approaches analyzed

Open problems and future directions identified

Abstract

The Eisenbud-Green-Harris (EGH) conjecture offers a generalization of the famous Macaulay's theorem about the Hilbert functions of homogeneous ideals in a polynomial ring $K[x_1,\ldots, x_n]$. In this survey paper, we provide a good compilation of results on the EGH conjecture that have been obtained so far. We discuss these results in terms of their approaches.