Avoidance Loci of Real Projective Varieties
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912719623421952 |
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| author | Pratt, Elizabeth Wang, Kexin |
| author_facet | Pratt, Elizabeth Wang, Kexin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15888 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Avoidance Loci of Real Projective Varieties Pratt, Elizabeth Wang, Kexin Algebraic Geometry 14P05, 14P25, 14M15 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. |
| title | Avoidance Loci of Real Projective Varieties |
| topic | Algebraic Geometry 14P05, 14P25, 14M15 |
| url | https://arxiv.org/abs/2511.15888 |