Skeletons and fans of logarithmic structures
Dan Abramovich, Qile Chen, Steffen Marcus, Martin Ulirsch, and, Jonathan Wise
TL;DR
This paper surveys various methods for generalizing fans in toric varieties within logarithmic geometry, highlighting their equivalences, differences, and applications.
Contribution
It provides a comprehensive overview of skeletons, Kato fans, Artin fans, and polyhedral cone complexes, clarifying their relationships and uses in logarithmic geometry.
Findings
Structures are equivalent under certain conditions
Different realizations serve diverse applications
Highlights current applications and future directions
Abstract
We survey a collection of closely related methods for generalizing fans of toric varieties, include skeletons, Kato fans, Artin fans, and polyhedral cone complexes, all of which apply in the wider context of logarithmic geometry. Under appropriate assumptions these structures are equivalent, but their different realizations have provided for surprisingly disparate uses. We highlight several current applications and suggest some future possibilities.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Commutative Algebra and Its Applications · Polynomial and algebraic computation