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Abstract. Let f : C → P n (C) be a holomorphic curves with hyperorder strictly less than 1, and algebraically nondegenerate over the field P 1 c which consists of c-periodic meromorphic functions on C. Let {Q j } q j=1 be fixed or c-periodic slowly moving hypersurfaces with degree d j (j ∈ {1, . . . , q}) in (weakly) N -subgeneral position in P n (C). In this paper, we prove a difference version of the second main theorem for f intersecting {Q j } q j=1 by using the Casorati determinant. A difference counterpart of the truncated second main theorem is also obtained. Our results extend the second main theorems for differences with fixed hyperplanes [9] or c-periodic slowly moving hyperplanes [10].