Arkiv för Matematik

Volume 56 (2018)

Number 1

Varieties of apolar subschemes of toric surfaces

Pages: 73 – 99



Matteo Gallet (Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria)

Kristian Ranestad (Department of Mathematics, University of Oslo, Norway)

Nelly Villamizar (Department of Mathematics, Swansea University, Swansea, United Kingdom)


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.


toric surfaces, apolarity, apolar schemes, powersum varieties

2010 Mathematics Subject Classification

14J99, 14M25, 14N99

N. Villamizar acknowledges the support of RICAM, Linz, where she developed part of the research contained in this paper. M. Gallet would like to thank Josef Schicho and Hamid Ahmadinezhad for helpful comments, especially about the introduction. 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).

Received 18 July 2016

Received revised 27 October 2016

Accepted 5 June 2017

Published 30 April 2018