Varieties of apolar subschemes of toric surfaces

Matteo Gallet Austrian Academy of Sciences Kristian Ranestad University of Oslo Nelly Villamizar Swansea University

Algebraic Geometry mathscidoc:1912.43006

Arkiv for Matematik, 56, (1), 73 – 99, 2018
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
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  title={Varieties of apolar subschemes of toric surfaces},
  author={Matteo Gallet, Kristian Ranestad, and Nelly Villamizar},
  booktitle={Arkiv for Matematik},
  pages={73 – 99},
Matteo Gallet, Kristian Ranestad, and Nelly Villamizar. Varieties of apolar subschemes of toric surfaces. 2018. Vol. 56. In Arkiv for Matematik. pp.73 – 99.
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