Our aim is to determine the tautological algebra generated by the cohomology classes of the Brill-Noether loci in the rational cohomology of the moduli stack $\mathcal{ U}_C(n,d)$ of semistable bundles of rank $n$ and degree $d$. We show that for a general smooth projective curve $C$ of genus $g\geq 2$, $d=2g-2$, the tautological algebra of $ \mathcal{ U}_C(2,2g-2)$ (resp. the moduli stack $\mathcal{ SU}_C(2,\mathcal{ L})$ of semistable bundles of rank $2$ and determinant $\mathcal{ L}$ with $\deg(\mathcal{ L})=2g-2$) is generated by the divisor classes (resp. the class of the Theta divisor $\Theta$). This is previously known in rank one situation, called the (classical) Porteous formula.
Categories: math.AG math.KT
