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Isotopies of real trigonal curves on Hirzebruch surfaces. (English. Russian original) Zbl 1027.14030

J. Math. Sci., New York 113, No. 6, 804-809 (2003); translation from Zap. Nauchn. Semin. POMI 267, 133-142 (2000).
From the introduction: V. A. Rokhlin [Russ. Math. Surv. 33, 85-98 (1978); translation from Usp. Mat. Nauk 33, No. 5(203), 77-89 (1978; Zbl 0437.14013)] defined rigid isotopy as a path in the space of nonsingular plane real algebraic curves of a given degree. More generally, a rigid isotopy is a path in the space of curves of a given class. At present, the rigid isotopy classification of real algebraic curves is known for nonsingular curves of degree \(m\leq 6\) on \(\mathbb{R}\mathbb{P}^2\), for nonsingular curves of bidegree \((m,1)\), \((m,2)\), and \((3,3)\) on a hyperboloid and an ellipsoid and for some of the simplest classes of singular curves on the same surfaces.
Main theorem: A nonsingular real algebraic curve of bidegree \((m,3)\) on the Hirzebruch surface \(\Sigma_1\) is determined up to rigid isotopy by its real scheme and type of complex conjugation.

MSC:

14P25 Topology of real algebraic varieties
14H51 Special divisors on curves (gonality, Brill-Noether theory)
14J25 Special surfaces
14H50 Plane and space curves

Citations:

Zbl 0437.14013
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