The minimum distance of sets of points and the minimum socle degree

Let $\mathbb K$ be a field of characteristic 0. Let $\Gamma\subset\mathbb P^n_{\mathbb K}$ be a reduced finite set of points, not all contained in a hyperplane. Let $hyp(\Gamma)$ be the maximum number of points of $\Gamma$ contained in any hyperplane, and let $d(\Gamma)=|\Gamma|-hyp(\Gamma)$. If $I\subset R=\mathbb K[x_0,...,x_n]$ is the ideal of $\Gamma$, then in \cite{t1} it is shown that for $n=2,3$, $d(\Gamma)$ has a lower bound expressed in terms of some shift in the graded minimal free resolution of $R/I$. In these notes we show that this behavior is true in general, for any $n\geq 2$: $d(\Gamma)\geq A_n$, where $A_n=\min\{a_i-n\}$ and $\oplus_i R(-a_i)$ is the last module in the graded minimal free resolution of $R/I$. 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

A commutative algebraic approach to the fitting problem

Given a finite set of points $\Gamma$ in $\mathbb P^{k-1}$ not all contained in a hyperplane, the "fitting problem" asks what is the maximum number $hyp(\Gamma)$ of these points that can fit in some hyperplane and what is (are) the equation(s) of such hyperplane(s). If $\Gamma$ has the property that any $k-1$ of its points span a hyperplane, then $hyp(\Gamma)=nil(I)+k-2$, where $nil(I)$ is the index of nilpotency of an ideal constructed from the homogeneous coordinates of the points of $\Gamma$. Note that in $\mathbb P^2$ any two points span a line, and we find that the maximum number of collinear points of any given set of points $\Gamma\subset\mathbb P^2$ equals the index of nilpotency of the corresponding ideal, plus one.Comment: 8 page

Overview of Neutron-Proton Pairing

The role of neutron-proton pairing correlations on the structure of nuclei along the $N=Z$ line is reviewed. Particular emphasis is placed on the competition between isovector ($T=1$) and isoscalar $(T=0$) pair fields. The expected properties of these systems, in terms of pairing collective motion, are assessed by different theoretical frameworks including schematic models, realistic Shell Model and mean field approaches. The results are contrasted with experimental data with the goal of establishing clear signals for the existence of neutron-proton ($np$) condensates. We will show that there is clear evidence for an isovector $np$ condensate as expected from isospin invariance. However, and contrary to early expectations, a condensate of deuteron-like pairs appears quite elusive and pairing collectivity in the $T=0$ channel may only show in the form of a phonon. Arguments are presented for the use of direct reactions, adding or removing an $np$ pair, as the most promising tool to provide a definite answer to this intriguing question.Comment: 89 pages, 59 figures. Accepted for publication in Progress in Particle and Nuclear Physics (ELSEVIER

Bounding invariants of fat points using a coding theory construction

Let Z \subseteq \proj{n} be a fat points scheme, and let $d(Z)$ be the minimum distance of the linear code constructed from $Z$. We show that $d(Z)$ imposes constraints (i.e., upper bounds) on some specific shifts in the graded minimal free resolution of $I_Z$, the defining ideal of $Z$. We investigate this relation in the case that the support of $Z$ is a complete intersection; when $Z$ is reduced and a complete intersection we give lower bounds for $d(Z)$ that improve upon known bounds.Comment: 18 pages, 1 figure; accepted in J. Pure Appl. Algebr