On the first infinitesimal neighborhood of a linear configuration of points in P 2
Abstract
We consider the following open questions. Fix a Hilbert function,
h
, that occurs for a reduced zero-dimensional subscheme of
P
2
. Among all subschemes,
X
, with Hilbert function
h
, what are the possible Hilbert functions and graded Betti numbers for the first infinitesimal neighborhood,
Z
, of
X
(i.e. the double point scheme supported on
X
)? Is there a minimum (
h
min
) and maximum (
h
max
) such function? The numerical information encoded in
h
translates to a {\it type vector}, which allows us to find unions of points on lines, called {\it linear configurations}, with Hilbert function
h
. We give necessary and sufficient conditions for the Hilbert function and graded Betti numbers of the first infinitesimal neighborhoods of {\it all} such linear configurations to be the same. Even for those
h
for which the Hilbert functions or graded Betti numbers of the resulting double point schemes are not uniquely determined, we give one (depending only on
h
) that does occur. We prove the existence of
h
max
, in general, and discuss
h
min
. Our methods include liaison techniques.