A holomorphic map from an open Riemann surface M into projective space induces a conformal metric on M by pulling back the Fubini-Study metric. The Gauss curvature of this induced metric measures how the map distorts the surface — concentrated curvature means the map compresses large regions into small images, and vice versa.
The paper derives a curvature estimate using jet differentials. Jet differentials are algebraic invariants of the derivatives of the map — not just the first derivative (the tangent map) but the second, third, and higher derivatives considered as algebraic objects. When the map has high ramification over a generic hypersurface of sufficiently high degree, the jet differentials constrain the curvature.
High ramification means the map's derivative vanishes to high order over the hypersurface — the surface wraps around the target many times near the ramification locus. This repetition creates geometric rigidity: a map that ramifies heavily over many points can't also have small curvature everywhere, because the ramification forces the metric to concentrate.
The sufficiency of the degree condition is the key technical point. The hypersurface needs to be generic (no special position) and of high enough degree (enough constraints). Below the degree threshold, counterexamples exist. Above it, the curvature estimate holds universally.
Curvature from combinatorics. The algebraic degree of a hypersurface determines the geometric curvature of a surface. The connection is the jet differentials — they convert algebraic ramification data into differential-geometric curvature bounds.