Gorenstein toric Schubert varieties in Grassmannians

Abstract

A partial flag variety is a smooth projective homogeneous variety admitting an action of a maximal torus $T$. Schubert varieties are $T$-invariant subvarieties of the partial flag varieties. We study toric Schubert varieties in Grassmannian varieties with respect to the action of the torus $T$. Indeed, we present an explicit description of the fan of a Gorenstein toric Schubert variety in a Grassmannian, and we prove that any Gorenstein toric Schubert variety in a Grassmannian variety is Fano.

Figures (1)

• Figure 1: The fan of $X_{2314}$ in $G/B$ or $G/P$

Theorems & Definitions (29)

• Theorem 1.1
• Example 2.1
• Proposition 2.2: see BL20Singular for the smoothness; see WooYong06Gorenstein for the Gorenstein condition; see TX23isomGrassmannianSchubert for the isomorphism classification
• Example 2.3
• Proposition 3.1
• Lemma 3.2: see HodgesLakshmibai22
• proof : Proof of Proposition \ref{['prop_classify_toric_Schubert']}
• Remark 3.3
• Proposition 3.4: see GK94Bott, LMP_Handbook
• Example 3.5