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We define two tau functions, $\tau$ and $\hat{\tau}$ , on moduli spaces ofspectral covers of $GL(n)$ Hitchin's systems. Analyzing the properties of$\tau$, we express the divisor class of the universal Hitchin's discriminant interms of standard generators of the rational Picard group of the moduli spacesof spectral covers with variable base. The function $\hat{\tau}$ is used tocompute the divisor of canonical 1-forms with multiple zeros.

Tau functions, Hodge classes and discriminant loci on moduli spaces of Hitchin's spectral covers