Nonsingular algebraic curves in ${\bf R}{\rm P}\sp 1\times{\bf R}{\rm P}\sp 1$

Abstract

We give some restrictions for the mutual position of the connected components of a nonsingular algebraic curve in the product space ${\mathbf {R}}{P^1} \times {\mathbf {R}}{P^1}$ of two real projective lines. We obtain our main theorem by calculating the Brown invariant of a certain quadratic form determined by the algebraic curve. Moreover, we consider a double covering of ${\mathbf {C}}{P^1} \times {\mathbf {C}}{P^1}$ branched along the complexification of our curve and antiholomorphic involutions that are the lifts of the complex conjugation.