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

Larson, Hannah K.

Canadian mathematical bulletin, Tome 66 (2023) no. 2, pp. 359-379

Résumé

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.

Détail

DOI : 10.4153/S0008439522000340

Mots-clés : Intersection theory, moduli of vector bundles, Chow rings