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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2310.18571 |
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| _version_ | 1866909241243074560 |
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| author | Goel, Gaurav Dhruv |
| author_facet | Goel, Gaurav Dhruv |
| contents | The space of all pencils of conics in the plane $\mathbb{P} V$ (where $\dim V = 3$) is a projective Grassmannian $\mathbb{G} (1, \mathbb{P} \mathrm{Sym}^2 V^*)$ and admits a natural $\mathrm{PGL}(V)$ action. It is a classical theorem that this action has exactly eight orbits, and in fact that the orbit of a pencil $\ell \subset \mathbb{P} \mathrm{Sym}^2 V^*$ is determined completely by its position with respect to the Veronese surface $X \subset \mathbb{P} \mathrm{Sym}^2V^*$ of rank 1 conics and its secant variety $S(X) \subset \mathbb{P} \mathrm{Sym}^2 V^*$, which is the cubic fourfold of rank 2 conics. In this paper, we present some geometric descriptions of these orbits. Then, using a mixture of direct enumerative techniques and some Chern class computations, we present a calculation of the classes of the orbit closures in the Chow ring of this Grassmannian (and consequently also of their degrees under the Plücker embedding $\mathbb{G} (1, \mathbb{P} \mathrm{Sym}^2 V^*)\hookrightarrow \mathbb{P} Λ^2 \mathrm{Sym}^2 V^*$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_18571 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Chow Ring Classes of $\mathrm{PGL}_3$ Orbit Closures in $\mathbb{G}(1, 5)$ Goel, Gaurav Dhruv Algebraic Geometry The space of all pencils of conics in the plane $\mathbb{P} V$ (where $\dim V = 3$) is a projective Grassmannian $\mathbb{G} (1, \mathbb{P} \mathrm{Sym}^2 V^*)$ and admits a natural $\mathrm{PGL}(V)$ action. It is a classical theorem that this action has exactly eight orbits, and in fact that the orbit of a pencil $\ell \subset \mathbb{P} \mathrm{Sym}^2 V^*$ is determined completely by its position with respect to the Veronese surface $X \subset \mathbb{P} \mathrm{Sym}^2V^*$ of rank 1 conics and its secant variety $S(X) \subset \mathbb{P} \mathrm{Sym}^2 V^*$, which is the cubic fourfold of rank 2 conics. In this paper, we present some geometric descriptions of these orbits. Then, using a mixture of direct enumerative techniques and some Chern class computations, we present a calculation of the classes of the orbit closures in the Chow ring of this Grassmannian (and consequently also of their degrees under the Plücker embedding $\mathbb{G} (1, \mathbb{P} \mathrm{Sym}^2 V^*)\hookrightarrow \mathbb{P} Λ^2 \mathrm{Sym}^2 V^*$). |
| title | The Chow Ring Classes of $\mathrm{PGL}_3$ Orbit Closures in $\mathbb{G}(1, 5)$ |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2310.18571 |