  We investigate the origin of the prompt and delayed emission observed in the
short GRB 090510. We use the broad-band data to test whether the most popular
theoretical models for gamma-ray burst emission can accommodate the
observations for this burst. We first attempt to explain the soft-to-hard
spectral evolution associated with the delayed onset of a GeV tail with the
hypothesis that the prompt burst and the high energy tail both originate from a
single process, namely synchrotron emission from internal shocks.
Considerations on the compactness of the source imply that the high-energy tail
should be produced in a late-emitted shell, characterized by a Lorentz factor
greater than the one generating the prompt burst. However, in this hypothesis,
the predicted evolution of the synchrotron peak frequency does not agree with
the observed soft-to-hard evolution. Given the difficulties of a
single-mechanism hypothesis, we test two alternative double-component
scenarios. In the first, the prompt burst is explained as synchrotron radiation
from internal shocks, and the high energy emission (up to about 1 s following
the trigger) as internal shock synchrotron-self-Compton. In the second
scenario, in view of its long duration (\sim 100 s), the high energy tail is
decoupled from the prompt burst and has an external shock origin. In this case,
we show that a reasonable choice of parameters does indeed exist to accommodate
the optical-to-GeV data, provided the Lorentz factor of the shocked shell is
sufficiently high. Finally, we attempt to explain the chromatic break observed
around \sim 1e3 s with a structured jet model. We find that this might be a
viable explanation, and that it lowers the high value of the burst energy
derived assuming isotropy, \sim 1e53 erg, below \sim 1e49 erg, more compatible
with the energetics from a binary merger progenitor.
