In Beauville's Complex Algebraic Surfaces, problem II.20 we are asked to show that an irreducible curve $C$ in a smooth surface $S$ becomes smooth after a finite number of blowups. He says that a way to do this is to show that the arithmetic genus decreases everytime we blowup $C$ at a singular point.
There is a proof of this in Barth, Hulek, Peters, Van de Ven's Compact Complex Surfaces using the following idea (I'll paraphrase):
Consider the case when $C$ has only one singular point $P$. Let $x,y$ be local coordinates for $S$ in a neighbourhood $U$ of $P$, so that the blowup is described as the subvariety of $U\times\Bbb{P}^1$ given by $sy=tx$.
We will assume that $y=0$ is the only tangent direction of $C$ at $P$ and that the multiplicity of $\widetilde{C}$ at the corresponding point is the same (in the other cases, the multiplicity clearly decreases). Then $C$ is described locally by the power series $$f=ay^m+\sum_{i+i> m}a_{ij}x^iy^j.$$
Restricting to the open set $V_s=\{s\neq 0\}$ and writing $t$ instead of $\frac{t}{s}$ for simplicity, we get that the strict transform $\widetilde{C}$ is given by $$\widetilde{f}=at^m+\sum_{i+j>m}a_{ij}x^{i+j-m}t^j$$
Then we observe that the order with which $\widetilde{f}(x,0)$ vanishes at the origin is strictly less than the order with which $f(x,0)$ does, which shows that the process will eventually terminate.
I understand all that, but I still can't see how this relates to the arithmetic genus $p_a=h^1(C,\mathcal{O}_C)$