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The purpose of this article is to extend the uniqueness results for the twodimensional Calder\'on problem to unbounded potentials on general geometricsettings. We prove that the Cauchy data sets for Schr\"odinger equationsuniquely determines potentials in $L^{p}$ for $p> 4/3$. In doing so, we firstrecover singularities of the potential, from which point a $L^2$-based methodof stationary phase can be applied. Both steps are done via constructions ofcomplex geometric optic solutions and Carleman estimates.

The Calder\'on problem in the $L^p$ framework on Riemann surfaces