On imaginary plane curves and Spin quotients of complex surfaces by complex conjugation

Abstract

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 $k>0$.