The critical locus is indeed generically irreducible

Let K be a field and let n, d > 1 be integers. We show that for all f in a Zariski-dense open subset of the space of degree-d endomorphisms of P^n defined over K, the critical locus of f is geometrically integral. This extends a theorem of Ingram–Ramadas–Silverman and is completely new in positive characteristic. Our construction is elementary, using only basic facts about polynomial rings and irreducible spaces. Time permitting, we will discuss the question of smoothness. All this is joint work with Max Weinreich.