Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Sachiko Matsuoka
Journal: Trans. Amer. Math. Soc. 324 (1991), 87-107
MSC: Primary 14P25; Secondary 57M99
MathSciNet review: 986695
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] V. I. Arnol′d, The situation of ovals of real plane algebraic curves, the involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms, Funkcional. Anal. i Priložen. 5 (1971), no. 3, 1–9 (Russian). MR 0286790
  • [2] Edgar H. Brown Jr., Generalizations of the Kervaire invariant, Ann. of Math. (2) 95 (1972), 368–383. MR 0293642
  • [3] D. A. Gudkov, The topology of real projective algebraic varieties, Uspehi Mat. Nauk 29 (1974), no. 4(178), 3–79 (Russian). Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ, II. MR 0399085
  • [4] Lucien Guillou and Alexis Marin, Une extension d’un théorème de Rohlin sur la signature, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 3, A95–A98. MR 0440566
  • [5] Axel Harnack, Ueber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876), no. 2, 189–198 (German). MR 1509883, 10.1007/BF01442458
  • [6] David Hilbert, Ueber die reellen Züge algebraischer Curven, Math. Ann. 38 (1891), no. 1, 115–138 (German). MR 1510666, 10.1007/BF01212696
  • [7] G. Ishikawa, Construction of extremal real correspondences of $ {{\mathbf{P}}^1}$ (preprint, 1986).
  • [8] Maurizio Letizia, Quotients by complex conjugation of nonsingular quadrics and cubics in 𝑃³_{𝐶} defined over 𝑅, Pacific J. Math. 110 (1984), no. 2, 307–314. MR 726490
  • [9] Alexis Marin, Quelques remarques sur les courbes algébriques planes réelles, Seminar on Real Algebraic Geometry (Paris, 1977/1978 and Paris, 1978/1979), Publ. Math. Univ. Paris VII, vol. 9, Univ. Paris VII, Paris, 1980, pp. 51–68 (French). MR 700836
  • [10] George Wilson, Hilbert’s sixteenth problem, Topology 17 (1978), no. 1, 53–73. MR 0498591
  • [11] V. I. Zvonilov, Complex topological characteristics of real algebraic curves on a surface, Funktsional. Anal. i Prilozhen. 16 (1982), no. 3, 56–57 (Russian). MR 674013

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 14P25, 57M99

Retrieve articles in all journals with MSC: 14P25, 57M99

Additional Information

Keywords: Algebraic curve, complex conjugation, signature, Brown invariant, branched double covering
Article copyright: © Copyright 1991 American Mathematical Society