Abstract
Powersum varieties, also called varieties of sums of powers, have provided examples of interesting relations between varieties since their first appearance in the 19th century. One of the most useful tools to study them is apolarity, a notion originally related to the action of differential operators on the polynomial ring. In this work, we make explicit how one can see apolarity in terms of the Cox ring of a variety. In this way, powersum varieties are a special case of varieties of apolar schemes; we explicitly describe examples of such varieties in the case of two toric surfaces, when the Cox ring is particularly well-behaved.
Funding Statement
N. Villamizar acknowledges the support of RICAM, Linz, where she developed part of the research contained in this paper. M. Gallet is supported by Austrian Science Fund (FWF): W1214-N15, Project DK9 and (FWF): P26607 and (FWF): P25652. K. Ranestad acknowledges funding from the Research Council of Norway (RNC grant 239015).
Acknowledgment
M. Gallet would like to thank Josef Schicho and Hamid Ahmadinezhad for helpful comments, especially about the introduction.
Citation
Matteo Gallet. Kristian Ranestad. Nelly Villamizar. "Varieties of apolar subschemes of toric surfaces." Ark. Mat. 56 (1) 73 - 99, April 2018. https://doi.org/10.4310/ARKIV.2018.v56.n1.a6
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