Quotients of flag varieties and their birational geometry

We compute the Chow quotient of the complete flag variety of subspaces of a four dimensional complex vector space, show that it is smooth and a Mori Dream Space, and describe in detail its birational geometry.

💡 Research Summary

This paper provides a comprehensive geometric study of the Chow quotient X of the complete flag variety F of subspaces of C^4 by the natural action of the maximal diagonal torus H in PGL(4). The main achievement is an explicit and concrete description of X, proving its smoothness and revealing its intricate birational geometry.

The authors’ strategy involves decomposing F into H-invariant affine charts, each equivariantly isomorphic to the Lie algebra n of lower triangular nilpotent matrices. They first compute the combinatorial quotient of this toric space n by H, which yields a smooth projective toric variety X_σ, obtained from P^3 via a sequence of toric blow-ups. These X_σ varieties appear as limits of GIT quotients of F and thus are targets of contractions from the Chow quotient X.

The core construction arises from studying the natural birational maps between the X_σ corresponding to different affine charts. These maps generate a finite subgroup W(τ) of the Cremona group of P^3, called the “tile group,” which is essentially the octahedral symmetry group S4 ⋊ Z/2Z. The authors then show that X is isomorphic to the “tile threefold,” a smooth variety constructed by performing a specific sequence of blow-ups on P^3 that resolves the indeterminacies of all elements in W(τ). This explicit construction proves that X is a smooth rational threefold with Picard number 12.

The paper then delves into the birational geometry of X. Its Picard group is generated by 12 “boundary divisors,” which parametrize codimension-one degenerations of the general H-orbit closure. The intersection theory is analyzed in terms of these divisors. A key result is that X is a weak Fano manifold, meaning its anticanonical divisor -K_X is nef and big. Consequently, X is a Mori Dream Space (MDS). This implies its Nef and Effective cones are polyhedral, and all its birational models can be understood via a finite number of Small Q-factorial Modifications (SQMs) and contractions.

The authors explicitly compute these cones and describe the contractions associated with their extremal rays. Notably, they identify contractions from X onto the Chow quotients of partial flag varieties of C^4, such as the Grassmannian Gr(2,4). Furthermore, they prove that the automorphism group of X is precisely the tile group W(τ), fully characterizing its symmetries.

In summary, this work offers the first complete geometric description of a non-trivial Chow quotient for a non-toric homogeneous space. It demonstrates how the quotient of a highly symmetric classical variety (the complete flag) inherits a different, rich type of symmetry (from the Weyl group) and exhibits a complex birational landscape, serving as a detailed case study at the intersection of geometric invariant theory, toric geometry, and the minimal model program.