This article continues the study of multiple blocking sets in PG(2, q). In [A. Blokhuis, L. Storme, T. Szonyi, Lacunary polynomials, multiple blocking sets and Baer subplanes. J. London Math. Soc. (2) 60 (1999), 321–332. MR1724814 (2000j:05025) Zbl 0940.51007], using lacunary polynomials, it was proven that t-fold blocking sets of PG(2, q), q square, t <q¼/2, of size smaller than t(q + 1) + cqq¿, with cq = 2-¿ when q is a power of 2 or 3 and cq = 1 otherwise, contain the union of t pairwise disjoint Baer subplanes when t = 2, or a line or a Baer subplane when t = 1. We now combine the method of lacunary polynomials with the use of algebraic curves to improve the known characterization results on multiple blocking sets and to prove a t (mod p) result on small t-fold blocking sets of PG(2, q = pn), p prime, n = 1.