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.