Rational surfaces with a non-arithmetic automorphism group

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-11-11 DOI:10.1112/blms.13175

Jennifer Li, Sebastián Torres

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Abstract

In [Algebraic surfaces and hyperbolic geometry, Cambridge University Press, Cambridge, 2012], Totaro gave examples of a K3 surface such that its automorphism group is not commensurable with an arithmetic group, answering a question of Mazur in Section 7 of [Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 14–50]. We give examples of rational surfaces with the same property. Our examples $Y$ are Looijenga pairs, that is, there is a connected singular nodal curve $D \subset Y$ such that $K_{Y} + D = 0$ .

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具有非算术自同构群的有理曲面

托塔罗在[代数曲面与双曲几何，剑桥大学出版社，剑桥，2012]中举例说明了K3曲面的自变群与算术群不可通约，回答了马祖尔在[Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 14-50]第7节中提出的问题。我们举例说明具有相同性质的有理曲面。我们的例子 Y $Y$ 是 Looijenga 对，即存在一条连通的奇异结点曲线 D ⊂ Y $D （子集 Y$ ），使得 K Y + D = 0 $K_{Y} + D = 0$ .+ D = 0$ .

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