  Several recent studies have considered the implications for astrophysics and
cosmology of some possible nonclassical properties of spacetime at the Planck
scale. The new effects, such as a Planck-scale-modified energy-momentum
(dispersion) relation, are often inferred from the analysis of some quantum
versions of Minkowski spacetime, and therefore the relevant estimates depend
heavily on the assumption that there could not be significant interplay between
Planck-scale and curvature effects. We here scrutinize this assumption, using
as guidance a quantum version of de Sitter spacetime with known Inonu-Wigner
contraction to a quantum Minkowski spacetime. And we show that, contrary to
common (but unsupported) beliefs, the interplay between Planck-scale and
curvature effects can be significant. Within our illustrative example, in the
Minkowski limit the quantum-geometry deformation parameter is indeed given by
the Planck scale, while in the de Sitter picture the parameter of quantization
of geometry depends both on the Planck scale and the curvature scalar. For the
much-studied case of Planck-scale effects that intervene in the observation of
gamma-ray bursts we can estimate the implications of "quantum spacetime
curvature" within robust simplifying assumptions. For cosmology at the present
stage of the development of the relevant mathematics one cannot go beyond
semiheuristic reasoning, and we here propose a candidate approximate
description of a quantum FRW geometry, obtained by patching together pieces
(with different spacetime curvature) of our quantum de Sitter. This
semiheuristic picture, in spite of its limitations, provides rather robust
evidence that in the early Universe the interplay between Planck-scale and
curvature effects could have been particularly significant.
