  It is proven that for any topological or analytical types of isolated
singular points of plane curves, there exists a non-real irreducible plane
algebraic curve of degree $d$ which goes through $d^2$ real distinct points and
has imaginary singular points of the given types. This result is used to
construct a series of examples of complex algebraic surfaces $X$ defined over
$\R$ whose quotients $Y=X/\conj$ by the complex conjugation $\conj$ are $Spin$
simply connected 4-manifolds with signature $16k$, for arbitrary integer $k>0$.
