Spectral data for a pair of matrices of order three and an action of the group $\mathrm{GL}(2,\mathbb Z)$

We consider the 10-dimensional complex space whose points are cubic curves on the projective complex plane with three marked points. The triples of marked points on the curve are defined up to equivalence of divisors. We construct a natural action of the group $\mathrm{GL}(2,\mathbb Z)$ on this space.