# On the fiber cone of monomial ideals

## TL;DR

This paper studies the fiber cone of monomial ideals in two variables, showing conditions under which it is a hypersurface, Cohen–Macaulay, and determining minimal generators of its defining ideal.

## Contribution

It characterizes the fiber cone of monomial ideals in two variables, identifying when it is a hypersurface or Cohen–Macaulay, and explicitly finds minimal generators of the defining ideal.

## Key findings

- Fiber cone of certain monomial ideals is a hypersurface.
- Fiber cone has positive depth in specific cases.
- Cohen–Macaulay property characterized by minimal generators of the defining ideal.

## Abstract

We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $I\subset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for interesting classes of height $2$ monomial ideals $I\subset K[x,y]$, which are generated by $4$ elements. For these classes of ideals we also show that $F(I)$ is Cohen--Macaulay if and only if the defining ideal $J$ of $F(I)$ is generated by at most 3 elements. In all the cases a minimal set of generators of $J$ is determined.