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An anticanonical perspective on G/P Schubert varieties

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arxiv 2506.18388 v1 pith:56PYQWLB submitted 2025-06-23 math.AG

Changzheng Li , Konstanze Rietsch , Mingzhi Yang This is my paper · ORCID
classification math.AG
keywords schubertvarietyfactorialwhenanticanonicalequivalentfanogorenstein
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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.

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  1. Quantum Schubert calculus for smooth Schubert divisors of $F\ell_n$

    math.AG 2025-09 conditional novelty 7.0 of 10

    The quantum cohomology of the smooth Schubert divisor X_{w0 s_{n-1}} in Fl_n is presented, with a quantum Chevalley formula and quantum Schubert polynomials identical to those of Fl_n.

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