Topological arrangement of curves of degree 6 on cubic surfaces in \Bbb R P³

1996/08/19 by Grigory Mikhalkin, G. Mikhalkin, Mikhalkin, G.
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #math.AG

paper · pdf · doi:10.48550/arxiv.math/9608207

arxiv created 1996/08/19 · openalex publication_date 1996/08/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A quadric in \R P³ cuts a curve of degree 6 on a cubic surface in \R P³. The papers classifies the nonsingular curves cut in this way on non-singular cubic surfaces up to homeomorphism. Two issues new in the study related to the first part of the 16th Hilbert problem appear in this classification. One is the distribution of the components of the curve between the components of the non-connected cubic surface which turns out to depend on the patterns of arrangements (see Theorem 1). The other is presence of positive genus among the components of the complement and genus-related restrictions (see Theorems 3 and 4).

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