We show that Weyl Fermi arcs are generically accompanied by a divergence of the surface Berry curvature scaling as $1/k^2$, where $k$ is the distance to a hot-line in the surface Brillouin zone that connects the projection of Weyl nodes with opposite chirality but which is distinct from the Fermi arc itself. Such surface Berry curvature appears whenever the bulk Weyl dispersion has a velocity tilt toward the surface of interest. This divergence is reflected in a variety of Berry curvature mediated effects that are readily accessible experimentally, and in particular leads to a surface Berry curvature dipole that grows linearly with the thickness of a slab of a Weyl semimetal material in the limit of long lifetime of surface states. This implies the emergence of a gigantic contribution to the non-linear Hall effect in such devices.
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