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

TL;DR

This paper proves that for smooth hypersurfaces in complex projective 3-space, certain jet bundles become big when the degree exceeds specific bounds, advancing understanding of their geometric properties.

Contribution

It improves previous bounds on the degrees for which the Demailly-Semple jet bundles are big, using algebraic calculations.

Findings

On $X_3$, $ ext{O}_{X_3}(1)$ is big for $d extgreater= 11$

On $X_4$, $ ext{O}_{X_4}(1)$ is big for $d extgreater= 10$

Enhances previous results by Diverio

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.