Abridged: The giant radio galaxy M 87 with its proximity, famous jet, and very massive black hole provides a unique opportunity to investigate the origin of very high energy (VHE; E>100 GeV) gamma-ray emission generated in relativistic outflows and the surroundings of super-massive black holes. M 87 has been established as a VHE gamma-ray emitter since 2006. The VHE gamma-ray emission displays strong variability on timescales as short as a day. In this paper, results from a joint VHE monitoring campaign on M 87 by the MAGIC and VERITAS instruments in 2010 are reported. During the campaign, a flare at VHE was detected triggering further observations at VHE (H.E.S.S.), X-rays (Chandra), and radio (43 GHz VLBA). The excellent sampling of the VHE gamma-ray light curve enables one to derive a precise temporal characterization of the flare: the single, isolated flare is well described by a two-sided exponential function with significantly different flux rise and decay times. While the overall variability pattern of the 2010 flare appears somewhat different from that of previous VHE flares in 2005 and 2008, they share very similar timescales (~day), peak fluxes (Phi(>0.35 TeV) ~= (1-3) x 10^-11 ph cm^-2 s^-1), and VHE spectra. 43 GHz VLBA radio observations of the inner jet regions indicate no enhanced flux in 2010 in contrast to observations in 2008, where an increase of the radio flux of the innermost core regions coincided with a VHE flare. On the other hand, Chandra X-ray observations taken ~3 days after the peak of the VHE gamma-ray emission reveal an enhanced flux from the core. The long-term (2001-2010) multi-wavelength light curve of M 87, spanning from radio to VHE and including data from HST, LT, VLA and EVN, is used to further investigate the origin of the VHE gamma-ray emission. No unique, common MWL signature of the three VHE flares has been identified.
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