Bertini theorems over finite fields (following Bjorn Poonen)

The classical Bertini theorems state that if a projective scheme over an infinite field has a certain property (smooth, geometrically irreducible, etc.), then its sufficiently general hyperplane section has that property too. It is well-known that these theorems can fail for finite fields. In this talk we discuss how the Bertini theorems can be adapted for schemes over finite fields.