Deformations of canonical triple covers

TL;DR

This paper studies how the canonical triple cover structure of certain smooth varieties of general type persists under deformations, especially when the target variety is a minimal degree variety, revealing unique higher-dimensional phenomena.

Contribution

It demonstrates that for specific high-dimensional varieties with canonical triple covers, the deformation preserves the triple cover structure, highlighting unique properties in the extremal case.

Findings

Deformations of the canonical map remain triple covers in specified cases.

The extremal case with minimal degree target varieties exhibits unique deformation behavior.

No lower-dimensional analogues exist for these phenomena.

Abstract

In this paper, we show that if $X$ is a smooth variety of general type of dimension $m \geq 3$ for which the canonical map induces a triple cover onto $Y$, where $Y$ is a projective bundle over $\mathbf P^1$ or onto a projective space or onto a quadric hypersurface, embedded by a complete linear series, then the general deformation of the canonical morphism of $X$ is again canonical and induces a triple cover. The extremal case when $Y$ is embedded as a variety of minimal degree is of interest, due to its appearance in numerous situations. This is especially interesting as well, since it has no lower dimensional analogues.