Mathematische Zeitschrift· 2026Q1
Integral points over number fields: a Clemens complex jigsaw puzzle
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- 2026year
Short summary
An asymptotic formula is derived for the number of integral points on a singular quartic del Pezzo surface over number fields, using the universal torsor method.
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Key points
- An asymptotic formula is derived for integral points on a singular quartic del Pezzo surface over number fields.
- The method utilizes the universal torsor approach and o-minimal structures.
- The analysis involves a complex Clemens structure with exponentially many polytopes.
- The volumes of these polytopes form a key component of the asymptotic formula.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract We prove an asymptotic formula for the number of integral points of bounded log anticanonical height on a singular quartic del Pezzo surface over arbitrary number fields, with respect to the largest admissible boundary divisor. The resulting Clemens complex is more complicated than usual, and leads to particularly interesting effective cone constants, associated with exponentially many polytopes whose volumes appear in the expected formula. Like a jigsaw puzzle, these polytopes fit together to one large polytope. The volume of this polytope appears in the asymptotic formula that we obtain using the universal torsor method via o-minimal structures.
The authors' abstract, as published at the source. Mathematische Zeitschrift, 2026 · DOI ↗
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Field: Geometry and Topology
Geometry and TopologyMathematics