  Context. The spatial distribution of molecules around starless cores is a
powerful tool for studying the physics and chemistry governing the earliest
stages of star formation. Aims. Our aim is to study the chemical
differentiation in starless cores to determine the influence of large-scale
effects on the spatial distribution of molecules within the cores. Furthermore,
we want to put observational constraints on the mechanisms responsible in
starless cores for the desorption of methanol from the surface of dust grains
where it is efficiently produced. Methods. We mapped methanol, CH3OH, and
cyclopropenylidene, c-C3H2, with the IRAM 30m telescope in the 3 mm band
towards six starless cores embedded in different environments, and in different
evolutionary stages. Furthermore, we searched for correlations among physical
properties of the cores and the methanol distribution. Results. From our maps
we can infer that the chemical segregation between CH3OH and c-C3H2 is driven
by uneven illumination from the interstellar radiation field (ISRF). The side
of the core that is more illuminated has more C atoms in the gas-phase and the
formation of carbon-chain molecules like c-C3H2 is enhanced. Instead, on the
side that is less exposed to the ISRF the C atoms are mostly locked in carbon
monoxide, CO, the precursor of methanol. Conclusions. We conclude that
large-scale effects have a direct impact on the chemical segregation that we
can observe at core scale. However, the non-thermal mechanisms responsible for
the desorption of methanol in starless cores do not show any dependency on the
H2 column density at the methanol peak.
