  In this letter, we report on dual-frequency European VLBI Network (EVN)
observations of the faintest and least luminous radio cores in Seyfert nuclei,
going to sub-mJy flux densities and radio luminosities around 10^19 W/Hz. We
detect radio emission from the nuclear region of four galaxies (NGC 4051, NGC
4388, NGC 4501, and NGC 5033), while one (NGC 5273) is undetected at the level
of ~100 microJy. The detected compact nuclei have rather different radio
properties: spectral indices range from steep (alpha>0.7) to slightly inverted
(alpha=-0.1), brightness temperatures vary from T_B=10^5 K to larger than 10^7
K and cores are either extended or unresolved, in one case accompanied by
lobe-like features (NGC 4051). In this sense, diverse underlying physical
mechanisms can be at work in these objects: jet-base or outflow solutions are
the most natural explanations in several cases; in the case of the undetected
NGC 5273 nucleus, the presence of an advection-dominated accretion flow (ADAF)
is consistent with the radio luminosity upper limit.
