In this paper, we give characterizations of the classical generalized quadrangles H(3, q 2) and H(4, q 2), embedded in PG(3, q 2) and PG(4, q 2), respectively. The intersection numbers with lines and planes characterize H(3, q 2), and H(4, q 2) is characterized by its intersection numbers with planes and solids. This result is then extended to characterize all Hermitian varieties in dimension at least 4 by their intersection numbers with planes and solids.
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