Canonical double covers of minimal rational surfaces and the non-existence of carpets

This article delves into the relation between the deformation theory of finite morphisms to projective space and the existence of ropes, embedded in projective space, with certain invariants. We focus on the case of canonical double covers X of a minimal rational surface Y, embedded in P-N by a complete linear series, and carpets on Y, canonically embedded in P-N. We prove that these canonical double covers always deform to double covers and that canonically embedded carpets on Y do not exist. This fact parallels the results known for hyperelliptic canonical morphisms of curves and canonical ribbons, and the results for K3 double covers of surfaces of minimal degree and Enriques surfaces and K3 carpets. That canonical double covers of minimal rational surfaces should deform to double covers is not a priori obvious, for the invariants of most of these surfaces lie on or above the Castelnuovo line; thus, in principle, deformations of such covers could have birational canonical maps. In fact, many canonical double covers of non-minimal rational surfaces do deform to birational canonical morphisms. We also map the region of the geography of surfaces of general type corresponding to the surfaces X and we compute the dimension of the irreducible moduli component containing [X]. In certain cases we exhibit some interesting moduli components parameterizing surfaces S with the same invariants as X but with birational canonical map, unlike X. (c) 2012 Elsevier Inc. All rights reserved.