  In an earlier paper (D. S. Keeler, D. Rogalski, and J. T. Stafford, ``Naive
noncommutative blowing up,'' Duke Math. J., 126 (2005), 491-546), we defined
and investigated the properties of the naive blowup of an integral projective
scheme X at a single closed point. In this paper we extend those results to the
case when one naively blows up X at any suitably generic zero-dimensional
subscheme Z. The resulting algebra A has a number of curious properties; for
example it is noetherian but never strongly noetherian and the point modules
are never parametrized by a projective scheme. This is despite the fact that
the category of torsion modules in the quotient category qgr A is equivalent to
the category of torsion coherent sheaves over X.
  These results are used in the companion paper ``A class of noncommutative
projective surfaces'' to prove that a large class of noncommutative surfaces
can be written as naive blowups.
