  Let $X$ be a compact Riemann surface $X$ of genus at--least two. Fix a
holomorphic line bundle $L$ over $X$. Let $\mathcal M$ be the moduli space of
Hitchin pairs $(E ,\phi\in H^0(End(E)\otimes L))$ over $X$ of rank $r$ and
fixed determinant of degree $d$. We prove that, for some numerical conditions,
$\mathcal M$ is irreducible, and that the isomorphism class of the variety
$\mathcal M$ uniquely determines the isomorphism class of the Riemann surface
$X$.
