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August, 2021 Examples of Surfaces with Canonical Map of Maximal Degree
Ching-Jui Lai, Sai-Kee Yeung
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Taiwanese J. Math. 25(4): 699-716 (August, 2021). DOI: 10.11650/tjm/210105


It was shown by Beauville that if the canonical map $\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then $\operatorname{deg}(\varphi_{|K_M|}) \leq 36$. The first example of a surface with canonical degree $36$ was found by the second author. In this article, we show that for any surface which is a degree four Galois étale cover of a fake projective plane $X$ with the largest possible automorphism group $\operatorname{Aut}(X) = C_7:C_3$ (the unique non-abelian group of order $21$), the base locus of the canonical map is finite, and we verify that $35$ of these surfaces have maximal canonical degree $36$. We also classify all smooth degree four Galois étale covers of fake projective planes, which give possible candidates for surfaces of canonical degree $36$. Finally, we also confirm in this paper the optimal upper bound of the canonical degree of smooth threefolds of general type with sufficiently large geometric genus, related to earlier work of Hacon and Cai.

Funding Statement

The first author was partially supported by the Ministry of Science and Technology in Taiwan with grant number MOST 107-2115-M-006-020. The second author was partially supported by a grant from the National Science Foundation.


It is a pleasure for the second author to thank Donald Cartwright for his help on Magma commands. The authors would like to express their appreciation and thankfulness to the referees for very helpful comments and suggestions on the paper. This work is partially done during the first author's visit at Research Institute of Mathematical Sciences in Kyoto, National Center of Theoretical Sciences and National Taiwan University in Taiwan, and the second author's visit of the Institute of Mathematics of the University of Hong Kong. The authors thank the warm hospitality of the institutes.


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Ching-Jui Lai. Sai-Kee Yeung. "Examples of Surfaces with Canonical Map of Maximal Degree." Taiwanese J. Math. 25 (4) 699 - 716, August, 2021.


Received: 15 September 2020; Revised: 2 November 2020; Accepted: 13 January 2021; Published: August, 2021
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.11650/tjm/210105

Primary: 14J25 , 14J29

Keywords: Canonical map , special surfaces , surfaces of general type

Rights: Copyright © 2021 The Mathematical Society of the Republic of China


Vol.25 • No. 4 • August, 2021
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