In press in Astronomy & Astrophysics. Uses aa LaTeX macros.We present maps at high spatial and spectral resolution in emission lines of C2H, c-C3H2, C4H, 12CO and C18O of the edge of the Horsehead nebula obtained with the Plateau de Bure Interferometer (PdBI). The edge of the Horsehead nebula is a one-dimensional Photo-Dissociation Region (PDR) viewed almost edge-on. All hydrocarbons are detected at high signal-to-noise ratio in the PDR where intense emission is seen both in the H2 ro-vibrational lines and in the PAH mid-infrared bands. C18O peaks farther away from the cloud edge. Our observations demonstrate that C2H, cC3H2 and C4H are present in UV-irradiated molecular gas, with abundances nearly as high as in dense, well shielded molecular cores. PDR models (i) need a large density gradient at the PDR edge to correctly reproduce the offset between the hydrocarbons and H2 peaks and (ii) fail to reproduce the hydrocarbon abundances. We propose that a new formation path of carbon chains, in addition to gas phase chemistry, should be considered in PDRs: because of intense UV-irradiation, large aromatic molecules and small carbon grains may fragment and feed the interstellar medium with small carbon clusters and molecules in significant amount.We are grateful to the IRAM staff at Plateau de Bure, Grenoble and Pico Veleta, Granada for competent help with the observations and data reduction. IRAM is supported by the INSU/CNRS (France), MPG (Germany) and IGN (Spain). We acknowledge funding by the French CNRS/PCMI program.Peer reviewed