Upper bounds for virtual dimensions of Seiberg-Witten moduli spaces

Given a closed four-manifold with $b_1=0$ and a prime number $p$, we prove that for any mod $p$ basic class, the virtual dimension of the Seiberg-Witten moduli space is bounded above by $2p-4$ under a mild condition on $b_2^+$. As an application, we obtain adjunction inequalities for embedded surfaces with negative self-intersection number.