Wednesday, January 12, 2022 4:30 PM
Vaughan McDonald

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.