Contracted ideals and the Gröbner fan of the rational normal curve

Abstract

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