On the canonical rings of covers of surfaces of minimal degree

Abstract

Let S be a regular surface of general type with at worst canonical singularities and with basepoint-free canonical system. Let X be its canonical image. It is well known that X must be a canonical surface or a minimal degree surface. The main result of the authors completely describes the number and degree of the generators of the canonical ring of S in the second case. More concretely, if r = deg(X) and n is the degree of the canonical map, then (1) if n = 2 and r = 1, the canonical ring is generated in degree 1, plus one generator in degree 4; (2) in the other cases, the canonical ring is generated in degree 1, plus r(n−2) generators in degree 2 and r −1 generators in degree 3. This result, together with previous results of Ciliberto and Green, describes when the canonical ring of S is generated in degree less than or equal to 2: X is not a surface of minimal degree other than the plane and, in this last case, n 6= 2. The authors also construct a series of non-trivial examples of the theorem and prove that some expected ones do not exist. Finally, the authors apply their results to Calabi-Yau threefolds, obtaining analogous results. The key point here is that, for a Calabi-Yau threefold, the general member of a big and base-point-free linear system is a surface of general type.