Abstract
Let X be a minimal complex surface of general type such that its image \(\Sigma \) via the canonical map \({\varphi }\) is a surface; we denote by d the degree of \({\varphi }\). In this expository work, first of all we recall the known possibilities for \(\Sigma \) and d when \({\varphi }\) is not birational, which are quite a few, and then we consider the question of producing concrete examples for all of them. We present the two main methods of construction of such examples and we give several instances of their application. We end the paper by outlining the state of the art on this topic and raising several questions.
A Ciro, maestro e amico.
Research partially supported by FCT/Portugal through UID/MAT/04459/2020 and by project PRIN 2017SSNZAW\(\_\)004 “Moduli Theory and Birational Classification” of Italian MIUR. The first author is a member of Centro de Análise Matemática, Geometria e Sistemas Dinâmicos of Técnico/ Universidade de Lisboa. The second author is a member of GNSAGA of INDAM
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Mendes Lopes, M., Pardini, R. (2023). On the Degree of the Canonical Map of a Surface of General Type. In: Dedieu, T., Flamini, F., Fontanari, C., Galati, C., Pardini, R. (eds) The Art of Doing Algebraic Geometry. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-11938-5_13
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