  Given a closed four-manifold with $b_1=0$ and a prime number $p$, we prove
that for any mod $p^r$ basic class, the virtual dimension of the Seiberg-Witten
moduli space is bounded above by $2r(p-1)-2$ under some conditions on $r$ and
$b_2^+$. As an application, we obtain adjunction inequalities for embedded
surfaces with negative self-intersection number.
