Nonsingular algebraic curves in 𝑅𝑃¹×𝑅𝑃¹

Matsuoka, Sachiko

doi:10.1090/S0002-9947-1991-0986695-1

Nonsingular algebraic curves in $\textbf {R}\textrm {P}^ 1\times \textbf {R}\textrm {P}^ 1$
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by Sachiko Matsuoka

Trans. Amer. Math. Soc. 324 (1991), 87-107

DOI: https://doi.org/10.1090/S0002-9947-1991-0986695-1

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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.

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