AbstractThis article continues the study of multiple blocking sets in PG(2,q). In [A. Blokhuis, L. Storme, T. Szőnyi, 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 thatt-fold blocking sets of PG(2,q),qsquare,t<q¼/2, of size smaller thant(q+ 1) +cqq⅔, withcq= 2−⅓whenqis a power of 2 or 3 andcq= 1 otherwise, contain the union oftpairwise disjoint Baer subplanes whent≥ 2, or a line or a Baer subplane whent= 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 at(modp) result on smallt-fold blocking sets of PG(2,q=pn),pprime,n≥ 1.