Abstract
It is an open problem to describe the Ext and Tor groups for two torus-invariant Weil divisors on a toric variety, using only the combinatorial data of the underlying objects from toric geometry. We will give a survey on this description for the case of two-dimensional cyclic quotient singularities, in particular how this description is related with the continued fraction associated to a cyclic quotient singularity. Furthermore, we will elaborate on the applications of these modules and expectations of how to generalize the results to higher dimensions, highlighted by examples.
Studying the above problems sparked software development in polymake and Singular, heavily using the interface between the systems. Examples are accompanied by code snippets to demonstrate the functionality added and to illustrate how one may approach similar problems in toric geometry using these software packages.
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Acknowledgements
The author is supported by the DFG (German research foundation) priority program SPP 1489 ‘Computeralgebra’ and the thematic program ‘Combinatorial Algebraic Geometry’ of The Fields Institute in Toronto.
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Kastner, L. (2017). Toric Ext and Tor in polymake and Singular: The Two-Dimensional Case and Beyond. In: Böckle, G., Decker, W., Malle, G. (eds) Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-70566-8_17
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