The symbolic generic initial system of almost linear point configurations in \(\mathbb P^2\) (Q289721)

Given a set of distinct points \(\{p_1,\dots, p_r\}\) of \(\mathbb{P}^2\), consider the homogeneous (or equimultiple) fat point subscheme \(Z = m(p_1 +\dots + p_r)\); its ideal \(I_Z \subset K[x, y, z]\) consists of functions vanishing to at least order \(m\) at each point. If \(I\) is the ideal of \(\{p_1,\dots , p_r\}\), then \(I_Z\) is equal to the \(m\)th symbolic power of \(I\), denoted \(I^{(m)}\).NEWLINENEWLINEThe author focus on the description of the limiting behavior of the Hilbert functions of the uniform fat point ideals \(\{I^{(m)}\}_{m}\) as \(m\) gets large and \(I\) is the ideal of a point configuration where all but one of the points lies on a single line.NEWLINENEWLINEIt is well known that the Hilbert function of an ideal and its generic initial ideal are equal. Thus, to describe the limiting behavior of the Hilbert functions of \(\{I^{(m)}\}_m\), the author studies the reverse lexicographic symbolic generic initial system \(\{\mathrm{gin} (I^{(m)}\}_m\) of \(I\) and describe its limiting shape. The limiting shape \(P\) of \(\{\mathrm{gin} (I^{(m)}\}_m\) is defined to be the limit \(\lim_{m\rightarrow \infty}1/m P_{\mathrm{gin}(I^{(m)})}\) where \(P_{\mathrm{gin} (I^{(m)})}\) denotes the Newton polytope of \(\mathrm{gin}(I^{(m)})\).NEWLINENEWLINEThe main result of this paper is the Theorem 1.1 that describes the limiting shape of \(\{\mathrm{gin} (I^{(m)}\}_m\) when \(I\) is an ideal of a point configuration where all but one of the points lies on a single line.NEWLINENEWLINEThe almost linear point configuration addressed by Theorem 1.1 may be viewed as one step more complex than the case where all points lie on a smooth conic.NEWLINENEWLINEAn other important result is Theorem 3.1. The author shows that when \(I\) is the ideal of such an almost linear point configuration, \(I^{(m)}\) is componentwise linear for infinitely many \(m\) and this means that the minimal free resolution of the ideal has a very simple form.NEWLINENEWLINESection 2 is devoted to the known results useful to prove the main result. In Section 3, the author proves results on componentwise linearity for individual fat point ideals. Section 4 uses these results to prove Theorem 1.1.