How to make log structures

We introduce the concept of a viable generically toroidal crossing (gtc) Deligne--Mumford stack \(Y\). This generalizes the concept of Gorenstein toroidal crossing space, which in turn generalizes that of a simple normal crossing scheme. On such a space \(Y\), we define by explicit construction a na...

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Bibliographic Details
Published in:arXiv.org 2024-02
Main Authors: Corti, Alessio, Ruddat, Helge
Format: Article
Language:English
Subjects:
Isomorphism
Online Access:Get full text
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Summary:We introduce the concept of a viable generically toroidal crossing (gtc) Deligne--Mumford stack \(Y\). This generalizes the concept of Gorenstein toroidal crossing space, which in turn generalizes that of a simple normal crossing scheme. On such a space \(Y\), we define by explicit construction a natural sheaf \(\mathcal{LS}_Y\), intrinsic to \(Y\). Our main theorem states that the set of nowhere vanishing sections \(\Gamma(Y,\mathcal{LS}_Y^\times)\) is canonically bijective to the set of isomorphism classes of log structures on \(Y\) over \(k^\dagger\) compatible with the gtc structure. The definition of \(\mathcal{LS}_Y\) by explicit construction permits the effective construction of log structures on \(Y\); it also enables logarithmic birational geometry, in particular the construction -- in some cases -- of resolutions of singular log structures. Our work generalizes Theorem 3.22 in GS06 and our proof follows closely the proof of that theorem as given in GS06.
ISSN:2331-8422