Chow Group of 1-cycles of the Moduli of Parabolic Bundles of Rank 2 Over a Curve

Sujoy Chakraborty

Published 2019-07-31Version 1

Let us fix a nonsingular projective curve $X$ of genus $g\geq 3$ over $\mathbb{C}$, and choose a point $x\in X$. Let $\mathcal{M}$ denote the moduli space of isomorphism classes of stable vector bundles of rank 2 and fixed determinant $\mathcal{O}_X(x)$ over $X$. Let us moreover fix $n$ distinct closed points $S=\{p_1,p_2,..., p_n\}$ over $X$, referred to as $\textit{parabolic points}$, and parabolic weights $(\alpha):= 0\leq \alpha_1 <\alpha_2<1 $ over the parabolic points. We also assume that the weights are generic. Let $\mathcal{M}_\alpha$ denote the moduli space of $S$-equivalence classes of parabolic stable vector bundles of rank 2 over $X$ of fixed determinant $\mathcal{O}_X(x)$. The Chow groups of these moduli spaces are interesting objects to study. I. Choe and J. Hwang showed that there is a canonical isomorphism $CH_1^{\mathbb{Q}}(\mathcal{M}) \cong CH_0^{\mathbb{Q}}(X)$. Here our aim is to find $CH_1^{\mathbb{Q}}(\mathcal{M}_{\alpha})$ for generic weights $\alpha$.