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

Abstract

A point $p$ on a smooth complex projective curve $C$ of genus $g>3$ is subcanonical if the divisor $(2g-2)p$ is canonical. In the moduli space of pointed curves, the subcanonical locus is described by pairs $(C,p)$ as above, and it consists of three irreducible components of dimension $2g-1$. Apart from the hyperelliptic component ${\mathcal{G}}_g^{hyp}$, the other components ${\mathcal{G}}_g^{odd}$ and ${\mathcal{G}}_g^{even}$ 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 of pairs $(C,p)$ such that $h^0(C,(g-1)p)$ is at least $r+1$ and it has the same parity as $r+1$. In particular, we provide a lower bound on their dimension, and we prove its sharpness for $r<4$. 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.

Accepted version. Sections 3 and 5 have been shortened. Some typos have been corrected. 17 pages. For copyright reasons, we note that the final publication is available at Springer via http://dx.doi.org/10.1007/s00209-014-1401-8

References in corpus (1)

• Connected components of the moduli spaces of Abelian differentials with prescribed singularities