An anticanonical perspective on G/P Schubert varieties

TL;DR

This paper investigates the Cartier class group, factoriality, and Fano properties of Schubert varieties in flag varieties, providing explicit formulas and characterizations, especially for simply-laced types.

Contribution

It introduces a natural basis for the Cartier class group of Schubert varieties and characterizes their factorial and Fano properties with explicit formulas.

Findings

A natural basis for the Cartier class group of Schubert varieties is described.

Conditions for Schubert varieties to be factorial or Fano are characterized.

Equivalence of factoriality, Q-factoriality, and Betti number conditions in simply-laced types is established.

Abstract

We describe a natural basis of the Cartier class group of an arbitrary Schubert variety $X_{w,P}$ in a flag variety $G/P$ of general Lie type. We then characterise when the Schubert variety is factorial/Fano, along with an explicit formula for the anticanonical line bundle in these cases. We also prove that, for Schubert varieties in simply-laced types (only), being factorial is equivalent to being $Q$-factorial, and is equivalent to the equality of the Betti numbers $b_2(X_{w,P})=b_{2\ell(w)-2}(X_{w,P})$. Finally, we give a convenient characterisation of when a simply-laced Schubert variety is Gorenstein and when it is Gorenstein Fano.