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Abstract We introduce adic tropicalizations for subschemes of toric varieties as limits of Gubler models associated to polyhedral covers of the ordinary tropicalization. Our main result shows that Huber's adic analytification of a subscheme of a toric variety is naturally isomorphic to the inverse limit of its adic tropicalizations, in the category of locally topologically ringed spaces. The key new technical idea underlying this theorem is cofinality of Gubler models, which we prove for projective schemes and also for more general compact analytic domains in closed subschemes of toric varieties. In addition, we introduce a ‐topology and structure sheaf on ordinary tropicalizations, and show that Berkovich analytifications are limits of ordinary tropicalizations in the category of topologically ringed topoi.
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A Schmidt–Nochka theorem for closed subschemes in subgeneral position
In previous work, the authors established a generalized version of Schmidt’s subspace theorem for closed subschemes in general position in terms of Seshadri constants.We extend our theorem to weighted sums involving closed subschemes in subgeneral position, providing a joint generalization of Schmidt’s theorem with seminal inequalities of Nochka.A key aspect of the proof is the use of a lower bound for Seshadri constants of intersections from algebraic geometry, as well as a generalized Chebyshev inequality.As an application, we extend inequalities of Nochka and Ru–Wong from hyperplanes in 𝑚-subgeneral position to hypersurfaces in 𝑚-subgeneral position in projective space, proving a sharp result in dimensions 2 and 3, and coming within a factor of 3/2 of a sharp inequality in all dimensions.We state analogous results in Nevanlinna theory generalizing the second main theorem and Nochka’s theorem (Cartan’s conjecture).
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- Award ID(s):
- 2302298
- PAR ID:
- 10628550
- Publisher / Repository:
- De Gruyter
- Date Published:
- Journal Name:
- Journal für die reine und angewandte Mathematik (Crelles Journal)
- Volume:
- 819
- ISSN:
- 0075-4102
- Page Range / eLocation ID:
- 205-229
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1515/crelle-2024-0085
