Comparison of fundamental group schemes of a projective variety and an ample hypersurface

Abstract

Let $X$ be a smooth projective variety defined over an algebraically closed field, and let $L$ be an ample line bundle over $X$. We prove that for any smooth hypersurface $D$ on $X$ in the complete linear system $|L^{\otimes d}|$, the inclusion map $D\hookrightarrow X$ induces an isomorphism of fundamental group schemes, provided $d$ is sufficiently large and $\dim X\ge 3$. If $\dim X=2$, and $d$ is sufficiently large, then the induced homomorphism of fundamental group schemes remains surjective. We give an example to show that the homomorphism of fundamental group schemes induced by the inclusion map of a reduced ample curve in a smooth projective surface is not surjective in general.