Most plane curves over finite fields are not blocking

A BSTRACT . A plane curve C ⊂ P 2 of degree d is called blocking if every F q -line in the plane meets C at some F q -point. We prove that the proportion of blocking curves is o (1) when d ≥ 2 q − 1 and q → ∞ . We also show that the same conclusion holds for smooth curves under the somewhat weaker condition d ≥ 3 p and d, q → ∞ . Moreover, the two events in which a random plane curve is smooth and is blocking, respectively, are shown to be asymptotically independent. Extending a classical result on the number of F q -roots of random polynomials, we find that the limiting distribution of the number of F q -points in the intersection of a random plane curve and a fixed F q -line is Poisson with mean 1 . We also present an explicit formula for the proportion of blocking curves involving statistics on the number of F q -points contained in a union of k lines for k = 1 , 2 , . . . , q 2 + q + 1 .