Two-dimensional directed site percolation is studied in systems directed along the x-axis and limited by a free surface at y=± Cx^k. Scaling considerations show that the surface is a relevant perturbation to the local critical behaviour when k<1/z where z=ν_∥/ν is the dynamical exponent. The tip-to-bulk order parameter correlation function is calculated in the mean-field approximation. The tip percolation probability and the fractal dimensions of critical clusters are obtained through Monte-Carlo simulations. The tip order parameter has a nonuniversal, C-dependent, scaling dimension in the marginal case, k=1/z, and displays a stretched exponential behaviour when the perturbation is relevant. The k-dependence of the fractal dimensions in the relevant case is in agreement with the results of a blob picture approach. Comment: 13 pages, Plain TeX file, epsf, 6 postscript-figures, minor corrections