Alexander-Hirschowitz Theorem: Fix $r \ge 2$ and $d \ge 2$, and consider the linear system $L = L^{(r )}_d(-\sum_{i=1}^n 2p_i)$ consisting of hypersurfaces of degree at most $d$ in $r$ variables that are singular at $n$ general points $\{p_i\}$.Then
(a) For $d = 2$, the linear system $L$ is special if and only if $2 \le n \le r$.
(b) For $d \ge 3$, the linear system $L$ is special if and only if the triple $(r,d,n)$ is one of the following: $(2, 4, 5), (3, 4, 9), (4, 4, 14), (4, 3, 7)$.
Thus if we consider degree $5$ hypersurfaces in $\mathbb{P}^3$ having singularity at $15$ general points, then $(n, d, r)$ is not as in the list of the above Theorem. Therefore by above theorem, $L$ is non special and therefore having expected dimension which is negative. In other words, there does not exist any degree $5$ hypersurface in $\mathbb{P}^3$ which is singular along $15$ general points. Please correct me if i have made any mistake in understanding the statement of the Theorem.
