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| Main Author: | |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2209.02443 |
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Table of Contents:
- We investigate the positivity and extension of invertible sheaves on group homogeneous spaces over coherent bases. Bypassing the failure of standard limit arguments and the classical Weil--Cartier correspondence, we develop a valuative divisor theory on locally coherent schemes. This establishes an exact correspondence between effective valuative divisors and rank-one reflexive sheaves, yielding a non-Noetherian Ramanujam--Samuel theorem. To homologically control special fibre degenerations, we study morphisms of (N)-type; these govern the descent of generically trivial invertible sheaves and establish the theorems of the cube and the square without smoothness hypotheses. Utilizing the Picard-admissibility of group actions, we construct ample invertible sheaves explicitly from one-codimensional orbit boundaries. This achieves the rigid extension of generic polarizations to integral models over Prüfer bases, structurally generalizing Raynaud's classical proof of his quasi-projectivity theorems.