Codimension One Decompositions and Chow Varieties

A presentation of a degree d form in n+1 variables as the sum of homogenous elements “essentially” involving n variables is called a codimension one decomposition. Codimension one decompositions are introduced and the related Waring Problem is stated and solved. Natural schemes describing the codimension one decompositions of a generic form are defined. Dimension and degree formulae for these schemes are derived when the number of summands is the minimal one; in the zero dimensional case the scheme is showed to be reduced. These results are obtained by studying the Chow variety ∆n,s of zero dimensional degree s cycles in P. In particular, an explicit formula for deg∆n,s is determined.