p-Rank and semi-stable reduction of curves |
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| Affiliation: | 1. School of Environmental Science and Engineering, School of Chemical Engineering, Guangdong Provincial Key Laboratory of Petrochemcial Pollution Processes and Control, Guangdong University of Petrochemical Technology, Maoming 525000, Guangdong, China;2. School of Metallurgy and Chemical Engineering, Jiangxi University of Science and Technology, Ganzhou 341000, Jiangxi, China;1. Waksman Institute of Microbiology, Rutgers, The State University of New Jersey, Piscataway, NJ 08854, USA;2. Department of Horticulture and Landscape Architecture, Purdue University, West Lafayette, IN 47907, USA;3. Department of Plant Biology and Pathology, Rutgers, The State University of New Jersey, Piscataway, NJ 08901, USA;4. Shanghai Center for Plant Stress Biology, Shanghai Institutes for Biological Sciences, Chinese Academy of Sciences, Shanghai 200032, China;1. Department of Diagnostic Radiology, University of Texas MD Anderson Cancer Center, Houston, TX 77030, USA;2. Department Imaging Physics, University of Texas MD Anderson Cancer Center, Houston, TX 77030, USA;3. Department Endocrine Neoplasia and Hormonal Disorders, University of Texas MD Anderson Cancer Center, Houston, TX 77030, USA;4. Department Surgical Oncology, University of Texas MD Anderson Cancer Center, Houston, TX 77030, USA;5. Department Pathology, University of Texas MD Anderson Cancer Center, Houston, TX 77030, USA;1. Freie Universität Berlin, Department of Earth Sciences, Physical Geography, Malteserstraße 74-100, Haus H, 12249 Berlin, Germany;2. Universität Leipzig, Institute for Geography, Johannisallee 19a, 04103 Leipzig, Germany;1. Graduate School of Science and Engineering, Ibaraki University, Hitachi, 316-8511, Japan;2. Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, 277-8561, Japan;3. Graduate School of Medical Life Science, Yokohama City University, Yokohama, 230-0045, Japan;4. Biomedical Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, 305-8566, Japan;5. AIST-UTokyo Advanced Operando-Measurement Technology Open Innovation Laboratory (OPERANDO-OIL), National Institute of Advanced Industrial Science and Technology (AIST), Kashiwa, 277-0882, Japan;1. The Second Clinical Medical College, Zhejiang Chinese Medical University, 310053, Hangzhou, China;2. Department of Radiology, Zhejiang Provincial People''s Hospital, Affiliated People''s Hospital of Hangzhou Medical College, 310013, Hangzhou, China;3. Department of Hepatobiliary and Pancreatic Surgery, The First Affiliated Hospital, Zhejiang University School of Medicine, Hangzhou, China;4. Zhejiang Provincial Key Laboratory of Pancreatic Disease, Hangzhou, China;5. Department of General Surgery, The Second Affiliated Hospital of Zhejiang University School of Medicine, Hangzhou, Zhejiang, China;6. Department of Nuclear Medicine, The Second Hospital of Zhejiang University School of Medicine, Hangzhou, China;7. Institute of Artificial Intelligence and Remote Imaging, Hangzhou Medical College, 310000, Hangzhou, China |
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| Abstract: | Let R be a discrete complete valuation ring, with field of fractions K, and with algebraically closed residue field k of characteristic p > 0. Let X be a germ of an R-curve at an ordinary double point. Consider a finite Galois covering f: Y → X, whose Galois group G is a p-group, such that Y is normal, and which is étale above Xk≔ x × rk. Asume that Y has a semi-stable model :→ Y over R, and let y be a closed point of Y. If the inertia subgroup I(y) at y is cyclic of order pn, we compute the p-rank of tf−1 (y) by using a result of Raynaud. In particular, we prove that this p-rank is bounded by pn −1. |
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