Let $\mathcal{N}$ be the moduli space of rank two holomorphic vector bundles with fixed determinant of degree one on a curve of genus two. In a classic paper from 1969, Narasimhan and Ramanan proved that $\mathcal{N}$ is isomorphic to a quadratic line complex, giving an identification of $\mathcal{N}$ with an intersection of two quadrics in $\mathbb{P}^5$. Their construction uses Hecke modifications of vector bundles to define a map from $\mathcal{N}$ to the Grassmannian of lines in the moduli space of bundles with trivial determinant, which they had shown to be isomorphic to $\mathbb{P}^3$. We generalize their construction to the nilpotent cones of the analogous moduli spaces of Higgs bundles, using Hecke modifications.

This is joint work with Dan Avritzer (UFMG).