40 research outputs found
A commutative algebraic approach to the fitting problem
Given a finite set of points in not all contained
in a hyperplane, the "fitting problem" asks what is the maximum number
of these points that can fit in some hyperplane and what is (are)
the equation(s) of such hyperplane(s). If has the property that any
of its points span a hyperplane, then , where
is the index of nilpotency of an ideal constructed from the
homogeneous coordinates of the points of . Note that in
any two points span a line, and we find that the maximum number of collinear
points of any given set of points equals the index
of nilpotency of the corresponding ideal, plus one.Comment: 8 page
The minimum distance of sets of points and the minimum socle degree
Let be a field of characteristic 0. Let be a reduced finite set of points, not all contained in a
hyperplane. Let be the maximum number of points of
contained in any hyperplane, and let . If
is the ideal of , then in \cite{t1}
it is shown that for , has a lower bound expressed in terms
of some shift in the graded minimal free resolution of . In these notes we
show that this behavior is true in general, for any : , where and is the last module in
the graded minimal free resolution of . In the end we also prove that this
bound is sharp for a whole class of examples due to Juan Migliore (\cite{m}).Comment: 11 page