Proportion of blocking curves in a pencil

A BSTRACT . Let L be a pencil of plane curves defined over F q with no F q -points in its base locus. We investigate the number of curves in L whose F q -points form a blocking set. When the degree of the pencil is allowed to grow with respect to q , we show that the geometric problem can be translated into a purely combinatorial problem about disjoint blocking sets. We also study the same problem when the degree of the pencil is fixed.