Hyperplane anti-Bertini embeddings over finite fields

Abstract

Baker asked, as recorded by Poonen, whether a fixed smooth quasiprojective variety over a finite field must have a smooth rational hyperplane section after every sufficiently high-dimensional linearly nondegenerate embedding. Poonen predicted a negative answer for every positive-dimensional variety. We prove this predicted negative answer for each prescribed variety: if $X$ is nonempty, smooth, quasiprojective, and of pure positive dimension over $\F_q$, then for every sufficiently large $N$ there is a locally closed embedding $X\hookrightarrow\PP^N_{\F_q}$ whose components remain linearly nondegenerate after arbitrary scalar extension, but whose every $\F_q$-rational hyperplane section is singular. The construction assigns one closed point of $X$ to each rational hyperplane and forces the pulled-back linear form to have zero first-order jet at that point.