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

A point p ∈ C on a smooth complex projective curve of genus g ≥ 3 is subcanonical
if the divisor (2g − 2)p is canonical. The subcanonical locus Gg ⊂ Mg,1 described by
pairs (C, p) as above has dimension 2g − 1 and consists of three irreducible components.
Apart from the hyperelliptic component Ghyp
g , the other components Godd
g and Geven
g depend
on the parity of h0(C, (g − 1)p), and their general points satisfy h0(C, (g − 1)p) = 1
and 2, respectively. In this paper, we study the subloci Grg
of pairs (C, p) in Gg such that
h0(C, (g − 1)p) ≥ r + 1 and h0 (C, (g − 1)p) ≡ r + 1 (mod 2). 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.