In this talk, we discuss the pathological phenomena of del Pezzo surfaces with only Du Val singularities in positive characteristic. More precisely, we determine all the surface for each of the following conditions;
(NL): the pair of the minimal resolutions and their exceptional divisors do not lift to the Witt ring,
(NB): all the anti-canonical divisors are singular,
(ND): there are no del Pezzo surfaces over the field of complex numbers with the same Dinkyn type and Picard rank.
(NK): the Kodaira vanishing theorem for ample Weil divisor fails.
We will mainly focus on the condition (NL). This is joint work with Masaru Nagaoka.

(This is a talk of Tokyo-Kyoto AG seminar on Zoom, and will be given in Japanese.)