can the arithmetic genus of a smooth projective surface be arbitrary?

Question

Can the arithmetic genus of a smooth projective surface be arbitrary integer ?

The surfaces in $\mathbb P^3$ always have nonegative $g_a$ (it's $0$ if $1 \leq d \leq 3$ and $\binom{d-1}{3}$ if $d \geq 4$), so I wonder whether $g_a$ can be other values, in particular less than $-1$.

Answer 1

If $X = C \times \mathbb{P}^1$ then $$ g_a(X) = \chi(\mathcal{O}_X) - 1 = (1 - g(C)) - 1 = -g(C), $$ so it can take arbitrary non-positive values.