  In this paper, we study pathologies of Du Val del Pezzo surfaces defined over
an algebraically closed field of positive characteristic by relating them to
their non-liftability to the ring of Witt vectors. More precisely, we
investigate 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, Picard rank, and anti-canonical degree, (NK): there
exists an ample $\mathbb{Z}$-divisor which violates the Kodaira vanishing
theorem for $\mathbb{Z}$-divisors, 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. As a result, for each of these conditions, we
determine all the Du Val del Pezzo surfaces which satisfy the given one.

Algebraic Geometry