On the deformations of canonical double covers of minimal rational surfaces

The purpose of this article is to study the deformations of smooth surfaces $X$ of general type whose canonical map is a finite, degree 2 morphism onto a minimal rational surface or onto $\mathbf F_1$, embedded in projective space by a very ample complete linear series. Among other things, we prove that any deformation of the canonical morphism of such surfaces $X$ is again a morphism of degree 2. A priori, this is not at all obvious, for the invariants $(p_g(X),c_1^2(X))$ of most of these surfaces lie on or above the Castelnuovo line; thus, in principle, a deformation of such $X$ could have a birational canonical map. We also map the region of the geography of surfaces of general type corresponding to the invariants of the surfaces $X$ and we compute the dimension of the irreducible moduli component containing $[X]$. In certain cases we exhibit some interesting moduli components parametrizing surfaces $S$ whose canonical map has different behavior but whose invariants are the same as the invariants of $X$. One of the interests of the article is that we prove the results about moduli spaces employing crucially techniques on deformation of morphisms. The key point or our arguments is the use of a criterion that requires only infinitesimal, cohomological information of the canonical morphism of $X$. As a by-product, we also prove the non-existence of "canonically" embedded multiple structures on minimal rational surfaces and on $\mathbf F_1$.