Intersections of Real Symmetric Hypersurfaces

TL;DR

This paper establishes a symmetric version of Bézout's theorem for complex symmetric hypersurfaces, classifies intersection orbit types in projective space, and derives restrictions on real intersection points.

Contribution

It introduces a symmetric Bézout's theorem, classifies intersection orbit types in projective plane, and provides restrictions on real points in symmetric intersections.

Findings

Symmetric orbit type determined by degrees for transverse intersections

Complete classification of orbit types in projective plane

Restrictions on the number of real points in symmetric intersections

Abstract

We prove a symmetric version of B\'ezout'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 $\mathbb{P}^3_{\mathbb{C}}$, with a similar restriction on real points.