On the depth of graded rings associated to lex-segment ideals in $K[x,y]$

TL;DR

This paper investigates the algebraic properties of lex-segment ideals in two-variable polynomial rings, establishing that their associated graded ring and fiber cone share the same depth, which advances understanding of their structural characteristics.

Contribution

It proves that for lex-segment ideals in $K[x,y]$, the depths of the associated graded ring and fiber cone are equal, a new insight into their algebraic structure.

Findings

Depths of associated graded ring and fiber cone are equal for lex-segment ideals.

Provides new understanding of the structural properties of lex-segment ideals.

Enhances algebraic theory related to polynomial rings in two variables.

Abstract

In this article, we show that the depths of the associated graded ring and fiber cone of a lex-segment ideal in $K[x,y]$ are equal.