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On the $\mathrm{PGL}_2$-equivariant intersection theory of $\mathrm{Gr}(2,4)$

Yuxuan Sun

Abstract

We determine the $\mathrm{PGL}_2$-equivariant Chow ring of $\mathrm{Gr}(2,4)^s$, the $\mathrm{PGL}_2$-stable locus of $\mathrm{Gr}(2,4)$, over any algebraically closed based field of characteristic not equal to 2 or 3. In the process, we demonstrate that the quotient stack $[\mathrm{Gr}(2,4)^s/\mathrm{PGL}_2]$ can be presented as the quotient of an open subset of $\mathbb{P}^1$ by a suitably chosen $S_4\leq \mathrm{PGL}_2$. We also discuss some apparent difficulties with computing the full $\mathrm{PGL}_2$-equivariant Chow ring of $\mathrm{Gr}(2,4)$.

On the $\mathrm{PGL}_2$-equivariant intersection theory of $\mathrm{Gr}(2,4)$

Abstract

We determine the -equivariant Chow ring of , the -stable locus of , over any algebraically closed based field of characteristic not equal to 2 or 3. In the process, we demonstrate that the quotient stack can be presented as the quotient of an open subset of by a suitably chosen . We also discuss some apparent difficulties with computing the full -equivariant Chow ring of .

Paper Structure

This paper contains 16 sections, 23 theorems, 156 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Challenges with calculating $A^*_{\mathrm{PGL}_2}(\mathrm{Gr}(2, 4))$
  4. Organization
  5. Basic definitions and results
  6. Results from equivariant intersection theory
  7. Results from geometric invariant theory for pencils of binary cubics
  8. Statement of main results
  9. Describing the S4-symmetry on $\mathrm{Gr}$(2,4)s
  10. $\mathrm{Stab}_{\mathop{\mathrm{\mathrm{PGL}}}\limits_2}(p)$ for $p\in \mathop{\mathrm{\mathrm{Gr}}}\limits(2,4)^s$
  11. The $S_4$-symmetry on $\mathrm{Gr}(2,4)^s$
  12. Proof of Theorem \ref{['main-thm-S4-symmetry-stable-locus']}
  13. Proof of Theorem \ref{['main-thm-Chow-ring-stable-locus']}
  14. Discussions and future directions
  15. Interpreting Theorem \ref{['main-thm-Chow-ring-stable-locus']}
  16. ...and 1 more sections

Key Result

Theorem 2.1

Suppose that the base field $k$ is of characteristic not equal to 2. Denote the adjoint representation of $\mathop{\mathrm{\mathrm{PGL}}}\limits_2(k)$ by $\mathfrak{sl}_2$ (notice that $\mathfrak{sl}_2 \cong \mathop{\mathrm{\mathrm{Sym}}}\limits^2 W$ as $PGL_2$-representations). Then, the $\mathop{\

Figures (1)

  • Figure 1: Inclusion relationships between closures of the non-stable $PGL_2$-orbits of $\mathop{\mathrm{\mathrm{Gr}}}\limits(2,4)$

Theorems & Definitions (48)

  • Theorem 2.1: Ve98, Theorem 1
  • Proposition 2.2
  • proof
  • Theorem 2.3: Ne81
  • Proposition 2.4: Wa83, cf. Wa98
  • proof
  • Remark 2.5
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • ...and 38 more