  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 $I$.
