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Aspects of the Segre variety S_{1,1,1}(2)

Ronald Shaw, Neil Gordon, Hans Havlicek

Abstract

We consider various aspects of the Segre variety S := S_{1,1,1}(2) in PG(7,2), whose stabilizer group G_S < GL(8, 2) has the structure N {\rtimes} Sym(3), where N := GL(2,2)\times GL(2,2)\times GL(2,2). In particular we prove that S determines a distinguished Z_3-subgroup Z < GL(8, 2) such that AZA^{-1} = Z, for all A in G_S, and in consequence S determines a G_S-invariant spread of 85 lines in PG(7,2). Furthermore we see that Segre varieties S_{1,1,1}(2) in PG(7,2) come along in triplets {S,S',S"} which share the same distinguished Z_3-subgroup Z < GL(8,2). We conclude by determining all fifteen G_S-invariant polynomial functions on PG(7,2) which have degree < 8, and their relation to the five G_S-orbits of points in PG(7,2).

Aspects of the Segre variety S_{1,1,1}(2)

Abstract

We consider various aspects of the Segre variety S := S_{1,1,1}(2) in PG(7,2), whose stabilizer group G_S < GL(8, 2) has the structure N {\rtimes} Sym(3), where N := GL(2,2)\times GL(2,2)\times GL(2,2). In particular we prove that S determines a distinguished Z_3-subgroup Z < GL(8, 2) such that AZA^{-1} = Z, for all A in G_S, and in consequence S determines a G_S-invariant spread of 85 lines in PG(7,2). Furthermore we see that Segre varieties S_{1,1,1}(2) in PG(7,2) come along in triplets {S,S',S"} which share the same distinguished Z_3-subgroup Z < GL(8,2). We conclude by determining all fifteen G_S-invariant polynomial functions on PG(7,2) which have degree < 8, and their relation to the five G_S-orbits of points in PG(7,2).

Paper Structure

This paper contains 18 sections, 11 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Segre variety $\mathcal{S}:=\mathcal{S}_{1,1,1}(2)$
  3. Some invariant attributes of the Segre variety $\mathcal{S}_{1,1,1}(2).$
  4. Some subgroups of $\mathcal{G}_{\mathcal{S}}$
  5. The subgroup $\mathcal{G}_{\mathcal{S}}^{0}=\mathcal{N}_{0}\rtimes\operatorname*{Sym}(3)$ of $\mathcal{G}_{\mathcal{S}}$
  6. Generators for the groups $\mathcal{G}_{\mathcal{S}},~\mathcal{G}_{\mathcal{S}}^{0},~\mathcal{G}_{\mathcal{B}}$
  7. The distinguished $Z_{3}$-subgroup $\mathcal{Z}<\operatorname*{GL}(8,2)$
  8. Orbits and triplets
  9. The five $\mathcal{G}_{S}$-orbits $\mathcal{O}_{1},\mathcal{O}_{2},\mathcal{O}_{3},\mathcal{O}_{4},\mathcal{O}_{5}$ of points
  10. The triplet $\{\mathcal{S},\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime}\}$ of Segre varieties $\mathcal{S}_{1,1,1}(2)$
  11. The $\mathcal{G}_{\mathcal{B}}$-orbits of points
  12. Invariant polynomials
  13. The $\mathcal{G}_{\mathcal{B}}$-invariant polynomials of degree $\leq4$
  14. The $\mathcal{G}_{\mathcal{S}}$-invariant polynomial functions of degree $<8$
  15. The $\mathcal{G}_{\mathcal{S}}$-invariant hyperbolic quadric $\mathcal{H}_{7}$
  16. ...and 3 more sections

Key Result

Lemma 1

(See ShawBBC20.) If $|\psi|$ is odd then $Q$ has polynomial degree $d$ if and only if (i) every $d$-flat is $\psi$-odd and (ii) there exists at least one $(d-1)$-flat which is $\psi$-even. (Here condition (i) $\Longrightarrow\deg Q\leq d,$ and condition (ii) $\Longrightarrow\deg Q\geq d.$)

Figures (1)

  • Figure 1: the '8-cube'

Theorems & Definitions (20)

  • Lemma 1
  • Lemma 2
  • Remark 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Theorem 7
  • Remark 8
  • Lemma 9
  • Theorem 10
  • ...and 10 more