# blow-up of multiplicity 2 point on a curve is smooth

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?

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 $$n \ge 4$$.
• 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? May 4, 2020 at 11:04
• 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. May 4, 2020 at 11:23