Bertini theorems over finite fields

Over an infinite field $k$, the classical Bertini smoothness theorem tells us if a subscheme $X \subseteq \BP_k^n$ is smooth, then its intersection $H\cap X$ with a generic hyperplane is also smooth. Unfortunately, the theorem fails over finite fields, as such a hyperplane may not even exist! A natural question then is: can you find a hypersurface of higher degree with smooth intersection? Poonen proved precisely this in his seminal paper, and even showed a positive density of hypersurfaces will have smooth intersection! This talk discusses Poonen’s “sieve-theoretic” proof method, which also has stronger consequences about oddities over finite fields, including the existence of “space-filling” and “space-avoiding” curves. Time permitting, we may discuss either Poonen’s arithmetic analogue of Bertini over $\Z$ or the finite field Bertini irreducibility theorem.