Bertini's Theorem

Question

Let $p_1,...,p_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m_1,...,m_n$ at $p_1,...,p_n$. I would like to know if the following variation of Bertini's theorem holds.

Suppose $|L|\neq\emptyset$. Is it true that a general element $H\in|L|$ is an hypersurface of degree $d$ such that $Sing(H) = \{p_1,...,p_n\}$ and $mult_{p_i}H = m_i$ for $i = 1,...,n$ ?

With multiplicity $m_i$ I mean that $p_i$ is an ordinary $m_i$-uple singular point.

Answer 1

As pointed out by Alex in his comment, this is in general not true.

For instance, consider the case $N=d=n=m=2$. Then $|L|$ is the linear system of plane curves of degree $2$ passing through $p_1$ and $p_2$ with multiplicity $2$. Of course there is only one such a curve, namely the line through $p_1$ and $p_2$ counted with multiplicity $2$. In particular, $|L|$ is not empty but no element in $|L|$ has ordinary singularities.

However, what you want is true under some additional hypotheses. For instance, the following result is proven in Shustin's paper Lower deformation of isolated hypersurface singularities, Proposition 10 p. 242.

For any pair $N$, $k$ of positive integers set $$M(N, \,k):=2 \cdot \binom{k+N}{N}, \quad M(N, \,2k-1):=\binom{k+N}{N} + \binom{k+N-1}{N}.$$ Then the following holds.

Proposition. Assume $$\sum_{i=1}^n \binom{m_i+N-1}{N} < M(N, \, d).$$ Then if $p_1, \ldots, p_n$ are general points in $\mathbb{P}^N$ there exists a hypersurface $L$ of degree $d$ having ordinary singularities of multiplicity $m_1, \ldots, m_n$ at the points $p_1, \ldots, p_n$ and which is smooth elsewhere. Consequently, the general element of $|L|$ has the same properties.