Intersections of Real Symmetric Hypersurfaces

2024/09/30 by Samuel Lidz, Lidz, Samuel, Zachary Lihn +3
Mathematics · #Mathematics and Applications #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2409.19929

Abstract

We prove a symmetric version of Bézout's theorem. More precisely, we show that the symmetric orbit type of a transverse intersection of complex symmetric hypersurfaces in projective space is determined by the degrees. In the projective plane, we fully classify the possible orbit types of such intersection loci using completely elementary methods. From this classification, we obtain strong restrictions on the number of real points in the intersection of real symmetric curves. We also provide a partial classification in ℙ³_ℂ, with a similar restriction on real points.

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