Subcanonical points on projective curves and triply periodic minimal surfaces in the Euclidean space

A point p on a smooth complex projective curve C of genus g ≥ 3 is subcanonical if the divisor (2g-2)p is canonical. The subcanonical locus in the moduli space M_{g,1} described by pairs (C,p) as above has dimension 2g-1 and consists of three irreducible components. Apart from the hyperelliptic component, the other two components depend on the parity of h^0(C,(g-1)p), and their general points satisfy h^0(C,(g-1)p)=1 and 2, respectively. In this paper, we study the subloci of pairs (C,p) such that h^0(C,(g-1)p) ≥ r+1 and h^0(C,(g-1)p) has the same parity as r+1, for some non-negative integer r. In particular, we provide a lower bound on their dimension, and we prove its sharpness for r ≤ 3. As an application, we further give an existence result for triply periodic minimal surfaces immersed in the 3-dimensional Euclidean space, completing a previous result of the second author.