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 {\em 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 $\Delta_{n,s}$ of zero dimensional degree $s$ cycles in $\PP^n$. In particular, an explicit formula for $\deg\Delta_{n,s}$ is determined.