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.



Bertini's Theorem
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](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=aa&paperid=1082&option_lang=eng)*, 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.



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> **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.
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