On the Demailly-Semple jet bundles of hypersurfaces in $\CP^3$

Abstract

Let $X$ be a smooth hypersurface of degree $d$ in $\CP^3$. By totally algebraic calculations, we prove that on the third Demailly-Semple jet bundle $X_3$ of $X$, the bundle $\ocal_{X_3}(1)$ is big for $d\geq 11$, and that on the fourth Demailly-Semple jet bundle $X_4$ of $X$, the bundle $\ocal_{X_4}(1)$ is big for $d\geq 10$, improving a recent result of Diverio.