Morphisms with Only Mild Singular Fibers and Bertini Theorems over Finite Fields

We compute the asymptotic density of non-singular hypersurfaces of bidegree (3, d) in P^n×P^1 with only mild singular fibers over P^1 over a finite field F_q for n = 1, 2. When n = 1 and char F_q > 2, this asymptotic density is ζP1(2)^−2. When n = 2 and char F_q > 3, it is bounded below and above by ζP1 (2)^−1 ζP1 (3/2)^−1 and ζP1(2)^−2 .Honors thesi