On the canonical rings of covers of surfaces of minimal degree

In one of the main results of this paper, we find the degrees of the generators of the canonical ring of a regular algebraic surface $X$ of general type defined over a field of characteristic $0$, under the hypothesis that the canonical divisor of $X$ determines a morphism $\varphi$ from $X$ to a surface of minimal degree $Y$. As a corollary of our results and results of Ciliberto and Green, we obtain a necessary and sufficient condition for the canonical ring of $X$ to be generated in degree less than or equal to $2$. We construct new examples of surfaces satisfying the hypothesis of our theorem and prove results which show that many a priori plausible examples cannot exist. Our methods are to exploit the $\mathcal{O}_{Y}$-algebra structure on $\varphi_{*}\mathcal{O}_{X}$. These methods have other applications, including those on Calabi-Yau threefolds. We prove new results on homogeneous rings associated to a polarized Calabi-Yau threefold and also prove some existence theorems for Calabi-Yau covers of threefolds of minimal degree. These have consequences towards constructing new examples of Calabi-Yau threefolds.