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

Abstract

A point \(p\in C\) on a smooth complex projective curve of genus \(g\ge 3\) is subcanonical if the divisor \((2g-2)p\) is canonical. The subcanonical locus \(\mathcal {G}_g\subset \mathcal {M}_{g,1}\) described by pairs \((C,p)\) as above has dimension \(2g-1\) and consists of three irreducible components. Apart from the hyperelliptic component \(\mathcal {G}_g^\mathrm{hyp }\), the other components \(\mathcal {G}_g^\mathrm{odd }\) and \(\mathcal {G}_g^\mathrm{even }\) 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 \(\mathcal {G}_g^{r}\) of pairs \((C,p)\) in \(\mathcal {G}_g\) such that \(h^0(C,(g-1)p)\ge r+1\) and \(h^0\left( C,(g-1)p\right) \equiv r+1\,(\mathrm{mod }\,2)\). In particular, we provide a lower bound on their dimension, and we prove its sharpness for \(r\le 3\). 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.