阳性特征的Du Val del Pezzo表面的病理学和可提升性
Pathologies and liftability of Du Val del Pezzo surfaces in positive characteristic
摘要：
本文研究了定义在正特征的代数闭域上的Du Val del Pezzo曲面的病态，并将其与Witt向量环的不可升性联系起来.更确切地说，我们研究了条件（NB）：所有反典范除子都是奇异的，（ND）：在复数域上不存在具有相同Dynkin型、Picard秩和反正则度的Du Val del Pezzo曲面，（NK）：存在一个充分的$\mathbb{Z}$ -因子，它违反了$\mathbb{Z}$ -因子的Kodaira消灭定理，以及（NL）：该对$(Y, E)$不提升到Witt向量环，其中$Y$是最小分辨率，而$E$是它的约化例外因子。因此，对于这些条件中的每一个，我们确定满足给定条件的所有Du Val del Pezzo曲面。
英文摘要：
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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