  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 explicitely describe examples
of such varieties in the case of two toric surfaces, when the Cox ring is
particularly well-behaved.
