An infinitesimal condition to smooth ropes

Abstract

In this article we give a condition, which holds in a very general setting, to smooth a rope, of any dimension, embedded in projective space. As a consequence of this we prove that canonically embedded carpets satisfying mild geometric conditions can be smoothed. Our condition for smoothing a rope \(\widetilde{Y}\) can be stated very transparently in terms of the cohomology class of a suitable first order infinitesimal deformation of a morphism ϕ associated to \(\widetilde{Y}\). In order to prove these results we find a sufficient condition, of independent interest, for a morphism ϕ from a smooth variety X to projective space, finite onto a smooth image, to be deformed to an embedding. Another application of this result on deformation of morphisms is the construction of smooth varieties in projective space with given invariants. We illustrate this by constructing canonically embedded surfaces with \(c_{1}^{2}=3p_{g}-7\) and deriving some interesting properties of their moduli spaces. The results of this article bear further evidence to the complexity of the moduli of surfaces of general type and its sharp contrast with the moduli of other objects such as curves or K3 surfaces.