CFT partition functions and moduli spaces of canonical curves

Conformal field theories (CFT) represent a framework of fruitful interplay between some of the most advanced topics in theoretical physics and algebraic geometry. In particular, the investigation of the CFT partition functions is closely related to the analysis of the correspondence between analytic and algebraic properties of closed Riemann surfaces. In the present thesis, some new aspects of this correspondence, in particular the ones arising in the CFTs associated to string and superstring theories, are considered. More precisely, the algebraic parameters, determining the canonical curve associated to a non-hyperelliptic Riemann surface, are explicitly computed in terms of Riemann theta functions, evaluated at generic points of the curve. Moreover, the techniques here introduced are applied to the analysis of the singular locus of the theta function, also considered with respect to the Andreotty-Mayer approach to the Schottky problem, and to the restriction of the Siegel's measure to the moduli space of canonical curves.