A containment result in $\mathbb{P}^n$ and the Chudnovsky conjecture

TL;DR

This paper proves a new containment relation for radical ideals of general points in projective space and confirms the Chudnovsky Conjecture for a broad class of point sets, advancing understanding in algebraic geometry.

Contribution

It establishes a novel containment result for symbolic and regular powers of ideals of general points in projective space, supporting the Chudnovsky Conjecture.

Findings

Proves $I^{(nm)} ext{ } ext{ extlangle} ext{ }I^{m}$ containment for ideals of general points.

Confirms the Chudnovsky Conjecture for very general points with size at least $2^n$ in $ ext{ } ext{ extlangle} ext{ } ext{ extlangle} ext{ } ext{in} ext{ } ext{ extlangle} ext{ } ext{projective space.

Provides a new tool for understanding symbolic powers of ideals in algebraic geometry.

Abstract

In the paper we prove the containment $I^{(nm)}\subset M^{(n-1)m}I^m$, for a radical ideal $I$ of $s$ general points in $\mathbb{P}^n$, where $s\geq 2^n$. As a corollary we get that the Chudnovsky Conjecture holds for a very general set of at least $2^n$ points in $\mathbb{P}^n$.