Pathologies and liftability of Du Val del Pezzo surfaces in positive characteristic

Tatsuro Kawakami; Masaru Nagaoka

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arXiv · 2008.07700

Pathologies and liftability of Du Val del Pezzo surfaces in positive characteristic

Tatsuro Kawakami·Masaru Nagaoka

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Abstract

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.

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Tatsuro Kawakami, Masaru Nagaoka. 2020-08-18. Pathologies and liftability of Du Val del Pezzo surfaces in positive characteristic. https://arxiv.org/abs/2008.07700

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