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In this paper, we study Du Val del Pezzo surfaces over an algebraically closed field of positive characteristic. Such a surface X may satisfy the condition (NB): all the anti-canonical divisors are singular, (ND): there are no Du Val del Pezzo surfaces over the field of complex numbers with the same Dynkin type and Picard rank, and (NL): the pair (Y, E) does not lift to the ring of Witt vectors, where Y is the minimal resolution and E is its reduced exceptional divisor. On one hand, we show that (ND) ⇒ (NL) ⇒ (NB). On the other hand, for each condition above, we determine all the Du Val del Pezzo surfaces with the given condition. We also give the list of the automorphism groups of Du Val del Pezzo surfaces satisfying one of them. Furthermore, following Ye's method of classification of Du Val del Pezzo surfaces in characteristic zero, we also classify Du Val del Pezzo surfaces of the Picard rank one in characteristic two or three. Finally, we show that if Du Val del Pezzo surfaces are Frobenius split, then they satisfy none of the conditions above.

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We show the Bogomolov-Sommese vanishing theorem holds for a log canonical projective surface not of log general type in large characteristic. As an application, we prove that a log resolution of a surface pair of log Kodaira dimension is less than or equal to zero lifts to characteristic zero in large characteristic. Moreover, we give an explicit bound on the characteristic unless the log Kodaira dimension is equal to zero.