Chudnovsky's Conjecture for very general points in $\mathbb{P}_k^{N}$

TL;DR

This paper proves Chudnovsky's conjecture for very general points in projective space over characteristic zero fields and for points on quadrics, also establishing it for large symbolic powers of ideals.

Contribution

It confirms Chudnovsky's conjecture in new cases, including very general points and points on quadrics, and for large symbolic powers of ideals.

Findings

Chudnovsky's conjecture holds for very general points in projective space.

The conjecture is valid for points lying on quadrics without field restrictions.

It is true for large symbolic powers of any homogeneous ideal.

Abstract

We prove a long-standing conjecture of Chudnovsky for very general and generic points in $\mathbb{P}_k^N$, where $k$ is an algebraically closed field of characteristic zero, and for any finite set of points lying on a quadric, without any assumptions on $k$. We also prove that for any homogeneous ideal $I$ in the homogeneous coordinate ring $R=k[x_0, \ldots, x_N]$, Chudnovsky's conjecture holds for large enough symbolic powers of $I$.