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This 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 that  t -fold blocking sets of PG(2,  q ),  q  square,  t  &lt;  q  ¼ /2, of size smaller than  t ( q  + 1) +  c   q   q  ⅔ , with  c   q   = 2 −⅓  when  q  is a power of 2 or 3 and  c   q   = 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  =  p   n  ),  p  prime,  n  ≥ 1.

On multiple blocking sets in Galois planes