Symbolic powers of the ideal of$n$ general points in $P^{n-1}$

Problem L of Fröberg–Lundqvist–Oneto–Shapiro asks for the difference between the Hilbert series of ordinary and symbolic powers of the ideal of general points in projective space. We solve this completely for $n$ general points of ${\text{PP}}^{n-1}$. Besides a closed formula for $$\text{HS}(S\mathrm{/}{I}^m)-\text{HS}(S\mathrm{/}{I}^{(m)}),$$ we determine all minimal monomial generators of $I^{(m)}$, and describe the symbolic Rees algebra. We also show that containment $I^{(m)}\subseteq I^r$ is detected solely by initial degrees. This gives the exact containment threshold, the Waldschmidt constant $\text{walpha}$, the resurgence $\text{Res}$, and the asymptotic resurgence $\text{aRes}$: $$\text{walpha}(I)=\frac{n}{n-1},\text{Res}(I)=\text{aRes}(I)=\frac{2(n-1)}{n}\mathrm{.}$$ We also take the first step beyond $n$ points: for $n+1$ general points of ${\text{PP}}^{n-1}$ — again a rigid, non-monomial configuration — we identify the defining quadrics, resolve the case $n=3$ completely (a complete intersection, with $J^{(m)}=J^m$ for all $m$ and resurgence $1$), and propose an exact Waldschmidt-constant formula $\text{walpha}=\frac{n+1}{n-1}$ for all $n$, verified computationally in every case we could check.