|Location:||MSRI: Simons Auditorium|
Over a finite field k, choose a random homogeneous polynomial f of degree d in k[x,y,z]. What is the probability that f defines a smooth planar curve?
Questions like this are related to Bertini Theorems over a finite field.
The first such Bertini Theorem over a finite field was proved by Bjorn Poonen, who showed that the (asymptotic) probability of smoothness may be realized as a product a of local probabilities.
Recent work of myself with Melanie Matchett Wood generalizes this result. I will discuss this collection of ideas, focusing primarily on examples.