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

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