On spectral cover equations in Simpson integrable systems

Following Simpson we consider the integrable system structure on the moduli spaces of Higgs bundles on a compact K"ahler manifold $X$. We propose a description of the corresponding spectral cover of $X$ as the fiberwise projective dual to a hypersurface in the projectivization $\mathbb{P}({\mathcal{T}}_X\oplus {\mathcal{O}}_X)$ of the tangent bundle ${\mathcal{T}}_X$ to $X$. The defining equation of the hypersurface dual to the Simpson spectral cover is explicitly constructed in terms of the Higgs fields.