Spectral curves and discrete Painlev\'e equations

It is well known that isomonodromic deformations admit a Hamiltonian description. These Hamiltonians appear as coefficients of the characteristic equations of their Lax matrices, which define spectral curves for linear systems of differential and difference systems. The characteristic equations in the case of the associated linear problems for various discrete Painlev\'e equations is biquadratic in the Painlev\'e variables. We show that the discrete isomonodromic deformations that define the discrete Painlev\'e equations may be succinctly described in terms of the characteristic equation of their Lax matrices.