Chudnovsky's conjecture for very general points in PkN

We prove a long-standing conjecture of Chudnovsky for very general and generic points in PkN, 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[x0,…,xN], Chudnovsky's conjecture holds for large enough symbolic powers of I.