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?
Best Answer
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 $n \ge 4$.