# Normal bundles of curves on hypersurfaces

## Meeting Details

Abstract: Let X be a smooth, projective manifold and let Y be a smooth, projective submanifold. The normal bundle $N_{Y/X}$ of $Y$ in $X$ controls the deformations of $Y$ in $X$ and plays a central role in geometry. Many questions of geometry and arithmetic ranging from problems of hyperbolicity to rational connectivity are closely related to properties of normal bundles of curves. In this talk, I will discuss several results with Eric Riedl on the positivity properties of normal bundles of curves on hypersurfaces. As a consequence, we show that a general Fano complete intersection has very free rational curves of the minimal possible degree in any characteristic. We also obtain lower bounds on genera of curves on very general surfaces in projective space. As a corollary, we resolve Demailly's long-standing conjecture that the very general quintic surface is algebraically hyperbolic.