Abstract
This article contains geometrical classification of all fibres in pencils of curves of genus two, which is essentially different from the numerical one given by Ogg ([11]) and Iitaka ([7]).
Given a family π:X→D of curves of genus two which is smooth overD′=D−{0}, we define a multivalued holomorphic mapT π fromD′ into the Siegel upper half plane of degree two, and three invariants called “monodromy”, “modulus point” and “degree”. We assert that the family π is completely determined byT π, and its singular fibre by these three invariants. Hence all types of fibres are classified by these invariants and we list them up in a table, which is the main part of this article.
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Partially supported by the Sakkokai Foundation.
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Namikawa, Y., Ueno, K. The complete classification of fibres in pencils of curves of genus two. Manuscripta Math 9, 143–186 (1973). https://doi.org/10.1007/BF01297652
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DOI: https://doi.org/10.1007/BF01297652