Fast radio bursts are extragalactic radio transient events lasting a few milliseconds with a ~Jy flux at ~1 GHz. We propose that these properties suggest a neutron star progenitor, and focus on coherent curvature radiation as the radiation mechanism. We study for which sets of parameters the emission can fulfil the observational constraints. Even if the emission is coherent, we find that self-absorption can limit the produced luminosities at low radio frequencies and that an efficient re-acceleration process is needed to balance the dramatic energy losses of the emitting particles. Self-absorption limits the luminosities at low radio frequency, while coherence favours steep optically thin spectra. Furthermore, the magnetic geometry must have a high degree of order to obtain coherent curvature emission. Particles emit photons along their velocity vectors, thereby greatly reducing the inverse Compton mechanism. In this case we predict that fast radio bursts emit most of their luminosities in the radio band and have no strong counterpart in any other frequency bands.
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