| This talk reports on a joint work with Gian Pietro Pirola about periodic surfaces, i.e. compact connected Riemann surfaces of genus g>2 admitting a conformal minimal immersion into a flat 3-dimensional real torus. By Generalized Weierstrass Representation, each periodic surface admists a certain spin structure with at least two global sections. Depending on the parity of the number of sections, we give a description of the locus of periodic surfaces in the moduli space M_g, and we prove the existence into any flat real 3-torus of countably many immersed periodic surfaces of genus g. These results are a byproduct of our analysis of the subcanonical locus in the moduli space M_{g,1}, which parameterizes complex projective curves C of genus g endowed with a point p such that (2g-2)p is a canonical divisor of C. In particular, we provide a lower bound on the dimension of the subloci having h^0(C,(g-1)p)>r and the same parity as r+1, and we prove its sharpness for low values of r. |