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The intersection theory of the moduli stack of vector bundles on $\mathbb {P}^1$

Part of: Cycles and subschemes Families, fibrations

Published online by Cambridge University Press:  07 July 2022

Hannah K. Larson
Hannah K. Larson*
Affiliation:
Department of Mathematics, Stanford University, 380 Jane Stanford Way, Stanford, CA 94305, USA
*
e-mail: hlarson@stanford.edu
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Abstract

We determine the integral Chow and cohomology rings of the moduli stack $\mathcal {B}_{r,d}$ of rank r, degree d vector bundles on $\mathbb {P}^1$-bundles. We work over a field k of arbitrary characteristic. We first show that the rational Chow ring $A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$ is a free $\mathbb {Q}$-algebra on $2r+1$ generators. The isomorphism class of this ring happens to be independent of d. Then, we prove that the integral Chow ring $A^*(\mathcal {B}_{r,d})$ is torsion-free and provide multiplicative generators for $A^*(\mathcal {B}_{r,d})$ as a subring of $A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$. From this description, we see that $A^*(\mathcal {B}_{r,d})$ is not finitely generated as a $\mathbb {Z}$-algebra. Finally, when $k = \mathbb {C}$, the cohomology ring of $\mathcal {B}_{r,d}$ is isomorphic to its Chow ring.

Keywords

Intersection theorymoduli of vector bundlesChow rings

MSC classification

Primary: 14C15: (Equivariant) Chow groups and rings; motives
Secondary: 14C17: Intersection theory, characteristic classes, intersection multiplicities 14D20: Algebraic moduli problems, moduli of vector bundles

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 2 , June 2023 , pp. 359 - 379
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

During the preparation of this article, the author was supported by the Hertz Foundation and NSF GRFP under Grant No. DGE-1656518.

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