This is a carefully written paper on the geometry of spectral curves and all order dispersive integrable system. After a summary of algebraic geometry, the authors review in the reconstruction of an isospectral Lax system from its semiclassical spectral curve which is time-independent. The techniques for this reconstruction are closely related to those developed by Krichever to produce the algebro-geometric solutions of the Zakharov-Shabat hierarchies. The authors propose a definition for a Tau function and a spinor kernel closely related to Baker-Akhiezer functions, where times parametrize slow (of order \(1/N\)) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of \(1/N\), where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when \(N\) is large. The large \(N\) limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. The authors check that their conjectural Tau function satisfies the Hirota equations to the first two orders, and they conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. They analyze its consequences, namely the possibility of reconstructing order by order in \(1/N\) an isomonodromic problem given by a Lax pair, and the relation between correlators, the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry.