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- Title
Approximable Algebras as Subalgebras of Section Rings.
- Authors
Maclean, Catriona
- Abstract
In [ 2 ], Huayi Chen introduced approximable graded algebras, which he uses to prove a Fujita-type theorem in the arithmetic setting, and asked if any such algebra is the graded ring of a big line bundle on a projective variety. This was proved to be false in [ 8 ]. Continuing the analysis started in [ 8 ], we show that while not every approximable graded algebra is a sub algebra of the section ring of a big line bundle, we can associate to any approximable graded algebra |$\textbf{B}$| a projective variety |$X(\textbf{B})$| and an infinite divisor |$D(\textbf{B}) =\sum _{i=1}^\infty a_i D_i$| with |$a_i\rightarrow 0$| such that |$\textbf{B}$| is a subalgebra of $$\begin{equation*} R(D(\textbf{B}))=\oplus_n H^0(X(\textbf{B}), n D(\textbf{B})).\end{equation*}$$ We also establish a partial converse to these results by showing that if an infinite divisor |$D=\sum _i a_iD_i$| converges in the space of numerical classes, then any full-dimensional sub-graded algebra of |$\oplus _mH^0(X, \lfloor mD \rfloor))$| is approximable.
- Subjects
DIVISOR theory; ALGEBRA; RING theory
- Publication
IMRN: International Mathematics Research Notices, 2022, Vol 2022, Issue 1, p171
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnaa065