Algebraic hyperbolicity of very general hypersurfaces in projective spaces

We prove that a very general sextic threefold in ${\mathbb{P}}^4$ is algebraically hyperbolic, settling the last open case and completing the classification of algebraic hyperbolicity for very general hypersurfaces in projective space. The proof follows the Coskun–Riedl scroll construction, with the moduli of stable maps and the logarithmic geometry of fibered surfaces as new ingredients.