[Paper Review] F-regularity of large Schubert varieties

Michel Brion, Jesper Funch Thomsen|ArXiv.org|Aug 13, 2004

Advanced Algebra and Geometry9 references4 citations

TL;DR

This paper establishes that large Schubert varieties in $G \times G$-equivariant projective embeddings of a connected reductive algebraic group $G$ are globally $F$-regular in positive characteristic and of globally $F$-regular type in characteristic zero. Using Frobenius splitting techniques and the global $F$-regularity of flag varieties, the authors prove these varieties are normal and Cohen-Macaulay, resolving a long-standing question about their singularities without requiring desingularizations.

ABSTRACT

Let G denote a connected reductive algebraic group over an algebraically closed field k and let X denote a projective G x G-equivariant embedding of G. The large Schubert varieties in X are the closures of the double cosets BgB, where B denotes a Borel subgroup of G, and g is in G. We prove that these varieties are globally F-regular in positive characteristic, resp. of globally F-regular type in characteristic 0. As a consequence, the large Schubert varieties are normal and

Motivation & Objective

• To establish the global $F$-regularity of large Schubert varieties in $G \times G$-equivariant projective embeddings of a reductive group $G$.
• To extend the known properties of global $F$-regularity—such as normality and Cohen-Macaulayness—from flag varieties to larger Schubert varieties.
• To provide a characteristic-free proof of the Cohen-Macaulay property for large Schubert varieties using Frobenius splitting and $F$-regularity techniques.
• To demonstrate that these varieties are of globally $F$-regular type in characteristic zero by reduction to positive characteristic.

Proposed method

• Leverages the Frobenius splitting properties of large Schubert varieties established in prior work [2] and [18].
• Applies the equivalence between global $F$-regularity and Frobenius splitting in positive characteristic, as developed in [20].
• Uses the global $F$-regularity of flag varieties and their Schubert varieties as a foundational input [12].
• Reduces the problem to the $p$-adic setting by constructing a $\mathbb{Z}$-form of the coordinate ring of $G$, then reducing modulo $p$.
• Constructs a $G_p \times G_p$-equivariant embedding $X_p$ of the group $G_p$ over $\overline{\mathbb{F}_p}$, showing that the coordinate ring $S_p$ is finitely generated and normal.
• Proves that the $B_p \times B_p$-orbit $B_p w_0 B_p$ maps open in $X_p$, implying $X_p$ is an equivariant embedding of $G_p$, and lifts this to the global $F$-regularity of the closures of double cosets $BgB$.

Experimental results

Research questions

• RQ1 Are large Schubert varieties in $G \times G$-equivariant embeddings globally $F$-regular in positive characteristic?
• RQ2 Do large Schubert varieties have rational singularities in characteristic zero, and can this be deduced from $F$-regularity?
• RQ3 Can the Cohen-Macaulay and normal properties of large Schubert varieties be established without desingularization?
• RQ4 Is the global $F$-regularity of flag varieties sufficient to imply the same for their larger counterparts via reduction techniques?
• RQ5 Can the $F$-regularity of the coordinate rings of large Schubert varieties be proven using Frobenius splitting and module-theoretic methods?

Key findings

• Large Schubert varieties in any $G \times G$-equivariant projective embedding of a connected reductive group $G$ are globally $F$-regular in positive characteristic.
• These varieties are of globally $F$-regular type in characteristic zero, meaning their singularities are well-behaved in a derived sense.
• The closures of double cosets $BgB$ are normal and Cohen-Macaulay, extending known results from flag varieties and canonical compactifications.
• The proof relies on reduction modulo $p$ and the construction of a $\mathbb{Z}$-form of the coordinate ring of $G$, ensuring the $F$-regularity lifts from positive to characteristic zero.
• The method avoids the need for desingularizations, providing a direct proof of the Cohen-Macaulay property via Frobenius splitting and $F$-regularity.
• The result confirms that large Schubert varieties in the space of $n \times n$ matrices are Cohen-Macaulay, consistent with earlier results by Knutson and Miller.