Contracted ideals and the Groebner fan of the rational normal curve

TL;DR

This paper explores the structure of contracted ideals in two variables and the Groebner fan of the rational normal curve, revealing deep connections and classifying Cohen-Macaulay cases.

Contribution

It provides a complete classification of Cohen-Macaulay contracted ideals and explicitly describes Cohen-Macaulay initial ideals of the rational normal curve's ideal.

Findings

Classified contracted ideals with Cohen-Macaulay associated graded rings.

Explicitly determined all Cohen-Macaulay initial ideals of the rational normal curve.

Revealed a surprising connection between contracted ideals and the Groebner fan.

Abstract

The paper has two goals: the study the associated graded ring of contracted homogeneous ideals in $K[x,y]$ and the study of the Groebner fan of the ideal $P$ of the rational normal curve in ${\bf P}^d$. These two problems are, quite surprisingly, very tightly related. We completely classify the contracted ideals with a Cohen-Macaulay associated graded rings in terms of the numerical invariants arising from Zariski's factorization. We determine explicitly all the initial ideals (monomial or not) of $P$ that are Cohen-Macaulay.