Avoidance Loci of Real Projective Varieties

TL;DR

This paper investigates the avoidance loci of real projective varieties, characterizing their structure, topology, and convexity properties, with explicit examples and bounds on connected components.

Contribution

It introduces the avoidance locus as a semi-algebraic set, generalizes the cone of positive polynomials, and analyzes its topological and convexity properties for various varieties.

Findings

Avoidance locus is an open semi-algebraic set.

Connected components are bounded by the topology of the real locus.

Avoidance loci are slice-convex.

Abstract

We study real linear spaces in projective space that avoid the real points of a non-degenerate projective variety. For a variety $X \subset \mathbb{P}^{n-1}$ with a real smooth point, we define the avoidance locus $\mathcal{A}_k(X)$ as the subset of the real Grassmannian $\mathrm{Gr}(k,n)_{\mathbb{R}}$ consisting of linear spaces that meet $X$ transversely but contain no real point of $X$. Our construction generalizes the cone of positive polynomials on $\mathbb{R}^n.$ We prove that the avoidance locus is an open semi-algebraic set equal to a union of regions in the complement of a higher Chow form, and that distinct regions are non-adjacent. We present explicit examples for linear spaces, curves, and surfaces, and provide bounds on the number of connected components of $\mathcal{A}_{n-1}(X)$ in terms of the topology of the real locus $X_{\mathbb{R}}$. Finally, we prove that avoidance loci are slice-convex.