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Singular del Pezzo surfaces and isotropic flag varieties

Michele Bianco, Luis E. Solá Conde

TL;DR

This work computes the normalized Chow quotient $X$ of the complete flag variety $F$ of isotropic flags in ${\mathbb C}^4$ with respect to a fixed skew form, showing $X$ is a singular del Pezzo surface of degree $4$ with two ${\rm A}_1$ singularities. The authors develop a framework where $X$ arises as the inverse limit of combinatorial quotients of torus-invariant affine charts, analyze a Cremona action of the Weyl group on a birational model $X'$, and prove $X\cong X'$; they then establish the anticanonical embedding of $X$ into ${\mathbb P}^4$ as a complete intersection of two quadrics and determine its Mori cone and automorphism group. The results reveal that $X$ is a Mori Dream Space with a rich boundary structure consisting of eight boundary divisors reflecting an eightfold subdivision of the fixed-point octagon. The paper thus demonstrates how Chow quotients serve as master spaces for orbit degenerations in flag varieties and yield explicit birational and geometric descriptions of singular del Pezzo surfaces in low rank. The interplay between torus combinatorics, BB-decomposition, and birational geometry provides a concrete instance of Chow quotients encoding boundary phenomena and birational contractions. Overall, the work contributes to the understanding of Chow quotients of general rational homogeneous manifolds and their birational models, with explicit computations in the symplectic ${\rm C}_2$ case.

Abstract

We compute the Chow quotient of the complete flag variety of isotropic subspaces of a four dimensional complex vector space with respect to a skew/symmetric form, and show that it is a singular del Pezzo surface of degree four.

Singular del Pezzo surfaces and isotropic flag varieties

TL;DR

This work computes the normalized Chow quotient of the complete flag variety of isotropic flags in with respect to a fixed skew form, showing is a singular del Pezzo surface of degree with two singularities. The authors develop a framework where arises as the inverse limit of combinatorial quotients of torus-invariant affine charts, analyze a Cremona action of the Weyl group on a birational model , and prove ; they then establish the anticanonical embedding of into as a complete intersection of two quadrics and determine its Mori cone and automorphism group. The results reveal that is a Mori Dream Space with a rich boundary structure consisting of eight boundary divisors reflecting an eightfold subdivision of the fixed-point octagon. The paper thus demonstrates how Chow quotients serve as master spaces for orbit degenerations in flag varieties and yield explicit birational and geometric descriptions of singular del Pezzo surfaces in low rank. The interplay between torus combinatorics, BB-decomposition, and birational geometry provides a concrete instance of Chow quotients encoding boundary phenomena and birational contractions. Overall, the work contributes to the understanding of Chow quotients of general rational homogeneous manifolds and their birational models, with explicit computations in the symplectic case.

Abstract

We compute the Chow quotient of the complete flag variety of isotropic subspaces of a four dimensional complex vector space with respect to a skew/symmetric form, and show that it is a singular del Pezzo surface of degree four.
Paper Structure (20 sections, 27 theorems, 67 equations, 10 figures, 2 tables)

This paper contains 20 sections, 27 theorems, 67 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Torus actions and associated quotients
  4. Polytopes of fixed points and sections.
  5. GIT, Chow and limit quotients
  6. Combinatorial quotients
  7. The complete flag of type ${\rm C}_2$
  8. Maximal torus action on the complete flag
  9. Fixed flags and their weights
  10. Fundamental subtori and their associated BB-cells
  11. Affine charts and their quotients
  12. The general $H$-orbit
  13. Combinatorial quotients of $H$-invariant affine charts
  14. The quotient fan and the BB-decomposition
  15. Boundary divisors
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $F$ be the complete flag variety of vector subspaces of ${\mathbb C}^4$, isotropic with respect to a nondegenerate skew-symmetric form, endowed with the natural action of the complex torus $H\subset\mathop{\rm PSp}\nolimits(4)$ of homothety classes of diagonal matrices. Then the normalized Chow

Figures (10)

  • Figure 1: Root system of type ${\rm C}_2$
  • Figure 2: The octagon $P$ decorated with the positive roots centered at $eB$.
  • Figure 3: Fixed point components of the ${\mathbb C}^*$-actions associated to $\mu_1$ and $\mu_2$.
  • Figure 4: Subdivisions of type $A$, $B$.
  • Figure 5: The quotient fan of ${\mathfrak n}$ by the action of $H$
  • ...and 5 more figures

Theorems & Definitions (70)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Proposition 2.7
  • Lemma 3.1
  • Remark 3.2
  • ...and 60 more