Vol. 311, No. 2, 2021

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Classification of smooth factorial affine surfaces of Kodaira dimension zero with trivial units

Tomasz Pełka and Paweł Raźny

Vol. 311 (2021), No. 2, 385–422

We give a corrected statement of (Gurjar and Miyanishi 1988, Theorem 2), which classifies smooth affine surfaces of Kodaira dimension zero, whose coordinate ring is factorial and has trivial units. Denote the class of such surfaces by 𝒮0. An infinite series of surfaces in 𝒮0, not listed in loc. cit., was recently obtained by Freudenburg, Kojima and Nagamine (2019) as affine modifications of the plane. We complete their list to a series containing arbitrarily high-dimensional families of pairwise nonisomorphic surfaces in 𝒮0. Moreover, we classify them up to a diffeomorphism, showing that each occurs as an interior of a 4-manifold whose boundary is an exceptional surgery on a 2-bridge knot. In particular, we show that 𝒮0 contains countably many pairwise nonhomeomorphic surfaces.

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affine surface, $\mathbb{C}^*$-fibration, log minimal model program, knot surgery, Kirby diagram
Mathematical Subject Classification
Primary: 14R05
Secondary: 14J26, 57M99, 57R65
Received: 26 February 2020
Revised: 15 December 2020
Accepted: 15 December 2020
Published: 31 July 2021
Tomasz Pełka
Basque Center for Applied Mathematics
Paweł Raźny
Instytut Matematyki
Uniwersytet Jagielloński