Let $f:X \to Y$ be a birational morphism between projective, irreducible surfaces. Assume $X$ is non-singular and $Y$ is a hypersurface in $\mathbb{P}^3$ (not necessarily smooth). Is the arithmetic genus of $X$ equal to that of $Y$?


More generally, given a smooth projective irreducible surface $X$, is it possible to find a smooth, projective hypersurface $Y$ in $\mathbb{P}^3$, birational to $X$?



Birational morphism and invariance of arithmetic genus
These are two different questions.


1) No. The arithmetic genus of a degree $d$ surface $Y\subset\mathbb{P}^3$ is $\chi (\mathcal{O}\_Y)-1=\binom{d-1}{3}$, regardless of the singularities of $Y$. If $Y$ has a $(d-1)$-uple point, it is rational, hence any desingularization $X$ of $Y$ has arithmetic genus $0$.


2) No. A smooth hypersurface $Y$ in $\mathbb{P}^3$ is simply connected, so any smooth surface birational to $Y$ must be simply connected.


