Nonsingular Algebraic Curves in RP1×RP1

We give some restrictions for the mutual position of the connected components of a nonsingular algebraic curve in the product space RP1 x RP1 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 CPI x CPI branched along the complexification of our curve and antiholomorphic involutions that are the lifts of the complex conjugation.