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On the Codimension-1 $\mathrm{PGL}_4$ Orbit Closures in $\mathrm{Gr}(2,10)$

Ari Krishna

Abstract

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 $[T]=12σ_1$.

On the Codimension-1 $\mathrm{PGL}_4$ Orbit Closures in $\mathrm{Gr}(2,10)$

Abstract

We study the natural action of on the Grassmannian , where and points of are pencils of quadrics in . Here while , so the generic orbit has codimension one and one expects a one-parameter family of generic orbits. We construct this family via the -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 -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 and geometric genus . Every divisorial fiber of the rational -map has class in . For the reduced codimension-one orbit closures one has for , , , and .

Paper Structure

This paper contains 14 sections, 18 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. The discriminant quartic and the j-map
  3. Intersection theory and the general fiber class
  4. Schubert calculus on the Grassmannian
  5. The complementary Schubert cycle
  6. The induced j-map on the Schubert slice
  7. The special fiber and the tangency divisor
  8. A normal form for the generic tangency pencil
  9. Codimension estimates for the bad repeated-root loci
  10. The structure of the special fiber
  11. The class of the nodal divisor
  12. Scheme-theoretic versus set-theoretic fibers
  13. Proof of the main theorem
  14. Further directions

Key Result

Theorem 1.1

Let $G=\mathrm{Gr}(2,\mathrm{Sym}^2 V^\vee)$ with $\dim V=4$, and let $\mathrm{PGL}(V)$ act naturally on $G$. Then:

Theorems & Definitions (41)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2: Simultaneous diagonalization
  • proof
  • Proposition 2.3: Orbit classification on the smooth locus
  • proof
  • Definition 2.4
  • Corollary 2.5
  • proof
  • Lemma 3.1
  • ...and 31 more