TY - JOUR
T1 - Pathologies and liftability of Du Val del Pezzo surfaces in positive characteristic
AU - Kawakami, Tatsuro
AU - Nagaoka, Masaru
N1 - Funding Information:
The authors would like to thank Professor Keiji Oguiso and Professor Shunsuke Takagi for their helpful advice and comments. They are indebted to Professor Hiroyuki Ito for helpful advice on rational quasi-elliptic surfaces. They would like to thank the referee for valuable advice which improved the paper. Discussions with Teppei Takamatsu, Yuya Matsumoto, and Takeru Fukuoka on the automorphism groups of Du Val del Pezzo surfaces have been insightful. They are grateful to Jakub Witaszek for letting them know about Remark . The authors would like to thank Fabio Bernasconi for telling them the paper []. They also wish to express their gratitude to Shou Yoshikawa, Yohsuke Matsuzawa, and Naoki Koseki for helpful discussions and comments. The authors are supported by JSPS KAKENHI Grant Numbers JP19J21085 and JP19J14397.
Funding Information:
The authors would like to thank Professor Keiji Oguiso and Professor Shunsuke Takagi for their helpful advice and comments. They are indebted to Professor Hiroyuki Ito for helpful advice on rational quasi-elliptic surfaces. They would like to thank the referee for valuable advice which improved the paper. Discussions with Teppei Takamatsu, Yuya Matsumoto, and Takeru Fukuoka on the automorphism groups of Du Val del Pezzo surfaces have been insightful. They are grateful to Jakub Witaszek for letting them know about Remark 2.10. The authors would like to thank Fabio Bernasconi for telling them the paper [2]. They also wish to express their gratitude to Shou Yoshikawa, Yohsuke Matsuzawa, and Naoki Koseki for helpful discussions and comments. The authors are supported by JSPS KAKENHI Grant Numbers JP19J21085 and JP19J14397.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2022/7
Y1 - 2022/7
N2 - 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 Z-divisor which violates the Kodaira vanishing theorem for 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.
AB - 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 Z-divisor which violates the Kodaira vanishing theorem for 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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U2 - 10.1007/s00209-022-02998-6
DO - 10.1007/s00209-022-02998-6
M3 - Article
AN - SCOPUS:85126269046
SN - 0025-5874
VL - 301
SP - 2975
EP - 3017
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
IS - 3
ER -