On the Codimension-1 $\mathrm{PGL}_4$ Orbit Closures in $\mathrm{Gr}(2,10)$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918417702846464 |
|---|---|
| author | Krishna, Ari |
| author_facet | Krishna, Ari |
| contents | We study the natural action of $\mathrm{PGL}(V)$ on the Grassmannian $G=\operatorname{Gr}(2,\operatorname{Sym}^2 V^\vee)$, where $\dim V=4$ and points of $G$ are pencils of quadrics in $\mathbb{P}(V)\cong \mathbb{P}^3$. Here $\dim G=16$ while $\dim \mathrm{PGL}(V)=15$, so the generic orbit has codimension one and one expects a one-parameter family of generic orbits. We construct this family via the $j$-invariant of the discriminant binary quartic of a pencil. We then determine the codimension-one orbit closures and compute their Chow classes. The smooth codimension-one orbit closures are the reduced fibers of the $j$-map on the smooth locus, while the unique boundary divisor is the closure of the orbit of a nodal quartic complete intersection of arithmetic genus $1$ and geometric genus $0$. Every divisorial fiber of the rational $j$-map has class $12σ_1$ in $A^1(G)$. For the reduced codimension-one orbit closures one has $[\overline{O_a}]=12σ_1$ for $a\neq 0,1728,\infty$, $[\overline{O_{1728}}]=6σ_1$, $[\overline{O_0}]=4σ_1$, and $[T]=12σ_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28619 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Codimension-1 $\mathrm{PGL}_4$ Orbit Closures in $\mathrm{Gr}(2,10)$ Krishna, Ari Algebraic Geometry 14L30 (Primary) 14M15, 14C15, 14C21 (Secondary) We study the natural action of $\mathrm{PGL}(V)$ on the Grassmannian $G=\operatorname{Gr}(2,\operatorname{Sym}^2 V^\vee)$, where $\dim V=4$ and points of $G$ are pencils of quadrics in $\mathbb{P}(V)\cong \mathbb{P}^3$. Here $\dim G=16$ while $\dim \mathrm{PGL}(V)=15$, so the generic orbit has codimension one and one expects a one-parameter family of generic orbits. We construct this family via the $j$-invariant of the discriminant binary quartic of a pencil. We then determine the codimension-one orbit closures and compute their Chow classes. The smooth codimension-one orbit closures are the reduced fibers of the $j$-map on the smooth locus, while the unique boundary divisor is the closure of the orbit of a nodal quartic complete intersection of arithmetic genus $1$ and geometric genus $0$. Every divisorial fiber of the rational $j$-map has class $12σ_1$ in $A^1(G)$. For the reduced codimension-one orbit closures one has $[\overline{O_a}]=12σ_1$ for $a\neq 0,1728,\infty$, $[\overline{O_{1728}}]=6σ_1$, $[\overline{O_0}]=4σ_1$, and $[T]=12σ_1$. |
| title | On the Codimension-1 $\mathrm{PGL}_4$ Orbit Closures in $\mathrm{Gr}(2,10)$ |
| topic | Algebraic Geometry 14L30 (Primary) 14M15, 14C15, 14C21 (Secondary) |
| url | https://arxiv.org/abs/2603.28619 |