The intersection theory of the moduli stack of vector bundles on ℙ¹

2021/04/29 by Larson, Hannah
#14C17 #14D20 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2104.14642

Abstract

We determine the integral Chow and cohomology rings of the moduli stack B_r,d of rank r, degree d vector bundles on ℙ¹ bundles. We first show that the rational Chow ring A_ℚ^*(B_r,d) is a free 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^*(B_r,d) is torsion-free and provide multiplicative generators for A^*(B_r,d) as a subring of A_ℚ^*(B_r,d). From this description, we see that A^*(B_r,d) is not finitely generated as a ℤ-algebra. Finally, the cohomology ring of B_r,d is isomorphic to its Chow ring.

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