On the symbolic powers of binomial edge ideals

TL;DR

This paper investigates conditions under which symbolic and ordinary powers of ideals coincide, demonstrating that for binomial edge ideals with quadratic Gr"obner bases, these powers are equal.

Contribution

It establishes a link between initial ideals and the equality of symbolic and ordinary powers, specifically applying this to binomial edge ideals with quadratic Gr"obner bases.

Findings

Symbolic and ordinary powers coincide for certain ideals.

The result applies to binomial edge ideals with quadratic Gr"obner bases.

Provides conditions under which ideal powers are equal.

Abstract

We show that under some conditions, if the initial ideal in$_<(I)$ of an ideal $I$ in a polynomial ring has the property that its symbolic and ordinary powers coincide, then the ideal $I$ shares the same property. We apply this result to prove the equality between symbolic and ordinary powers for binomial edge ideals with quadratic Gr\"obner basis.