The presence of symmetric and more generally -jet differentials on surfaces of general type play an important role in constraining the presence of entire curves (nonconstant holomorphic maps from to ). Green-Griffiths-Lang conjecture and Kobayashi conjecture are the pillars of the theory of constraints on the existence of entire curves on varieties of general type.

When the surface as a low ratio a simple application of Riemann-Roch is unable to guarantee abundance of symmetric or -jet differentials.

This talk gives an approach to show abundance on resolutions of hypersurfaces in with singularities and of low degree (low ). A new ingredient from a recent work with my student Michael Weiss gives that there are such hypersurfaces of degree (and potentially ). The best known result till date was degree .