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Suppose we have a smooth hypersurface $S \subset \mathbb{P}^3$ and a curve $C \subset S$ that is singular at $P \in C$ with multiplicity $\mu_P(C) = 2$ and not singular anywhere else. Then if let $\widetilde{C}$ be the blow-up of the curve $C$ at $P$, is $\widetilde{C}$ smooth?

From Corollary V.3.7 of Hartshorne we get that $p_a(C) - p_a(\tilde{C})= 1$, so the arithmetic genus decreases by one, does this mean that the multiplicity of $P$ also decreases by one?

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No, it is not. For instance, take $S$ to be a plane and $C$ a hypercusp, written in local coordinates as $$ y^2 = x^n. $$ A simple computation shows that the blowup of $C$ at the origin is given by $$ {y'}^2 = x^{n-2}, $$ hence it is singular if $n \ge 4$.

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  • $\begingroup$ Thanks, so as a follow-up question: if C is written in local coordinates with as highest degree terms at most degree 3, is the blowup then nonsingular? For example if we take C to be the intersection of S with its first polar at P, then S contains at most degree 3 terms and the first polar at most degree 2 terms, so is $\widetilde{C}$ then smooth? $\endgroup$
    – sebarev
    May 4, 2020 at 11:04
  • $\begingroup$ It is unclear to me what do you mean by the "degree of a highest degree term". Definitely, if the singularity of the curve is a node or a simple cusp, then $\tilde{C}$ is smooth. $\endgroup$
    – Sasha
    May 4, 2020 at 11:23

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