  We investigate the algebra and geometry of the independence conditions on
discrete random variables in which we fix some random variables and study the
complete independence of some subcollections. We interpret such independence
conditions on the random variables as an ideal of algebraic relations. After a
change of variables, this ideal is generated by generalized 2x2 minors of
multi-way tables and linear forms. In particular, let Delta be a simplicial
complex on some random variables and A be the table corresponding to the
product of those random variables. If A is Delta-independent table then A can
be written as the entrywise sum B+C where B is a completely independent table
and C is identically 0 in its Delta-margins.
  We compute the isolated components of the original ideal, showing that there
is only one component that could correspond to probability distributions, and
relate the algebra and geometry of the main component to that of the Segre
embedding. If Delta has fewer than three facets, we are able to compute
generators for the main component, show that it is Cohen--Macaulay, and give a
full primary decomposition of the original ideal.
