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People usually say that the quantum cohomology of $ \mathbb{P}^1$ Frobenius manifold $ QH^*(\mathbb{P}^1)$ , corresponds to dispersionless extended Toda hierarchy (e.g. page 6 of https://arxiv.org/pdf/math/0308152.pdf). I’m trying to understand this correspondence in more detail but I’m some having troubles. More specifically how do I recover the $ \tau$ -function? Should I be able toRead more

Let $ C$ an irreducible, integer and proper curve over $ k$ and $ f: C \to \mathbb{P}^1$ a morphism. Let assump that one has already proved that for every closed point $ b \in \mathbb{P}^1$ there exist a unique $ c_b$ with $ c_b = f^{-1}(b)$ (so $ f$ is bijective on closed points).Read more