The alpha problem \& line count configurations (Q402674)

Given a homogeneous ideal \(I\) in the polynomial ring \(R = k[{\mathbb P}^n] = k[x_0,\dots, x_n]\) over the field \(k\), the \(m\)th symbolic power is defined to be \(I^{(m)} = R \cap (\bigcap_{P} (I^m)_P )\) where the second intersection is taken over all associated primes \(P\) of \(I\) and the intersections are taken in the field of fractions of \(R\). It is well-known that \(I^{(nm)}\subseteq I^m\) for any \(m\geq 1\) (see [\textit{L. Ein} et al., Invent. Math. 144, No. 2, 241--252 (2001; Zbl 1076.13501); \textit{M. Hochster} and \textit{C. Huneke}, Invent. Math. 147, No. 2, 349--369 (2002; Zbl 1061.13005)]).NEWLINENEWLINEWhen \(I\) is an ideal of a finite set of distinct points, the symbolic power \(I^{(m)}\) defines a fat point scheme.NEWLINENEWLINE\textit{B. Harbourne} and \textit{C. Huneke} [J. Ramanujan Math. Soc. 28A, 247--266 (2013; Zbl 1296.13018)] conjectured several relationships for fat points, including the following:NEWLINENEWLINE{ Conjecture 1.2}. Let \(I\subset R = k[x_0,\dots, x_n]\) be the ideal of a finite set of distinct points in \({\mathbb P}^n\) and \(M = (x_0,\dots , x_n)\subset R\) be the irrelevant ideal. Then (i) \(I^{(rn)}\subseteq M^{r(n-1)}I^r\) holds for all \(r> 0\); (ii) \(I^{(rn-(n-1))}\subseteq M^{(r-1)(n-1)}I^r\) holds for all \(r > 0\).NEWLINENEWLINEIn this paper, the authors consider fat points supported on line count configurations, since they have special tools to work with. They observe that if Conjecture 1.2 is true, then we have an immediate relationship between the initial degrees in which there are non-zero forms in the symbolic power \(I^{(rn)}\), and the regular powers \(I^r\) and \(M^{r(n-1)}\) for part (i), and in the symbolic power \(I^{(rn-(n-1))}\) and the regular powers \(I^r\) and \(M^{(r-1)(n-1)}\) in part (ii). They state conjectures that give a relationship between the initial degrees of forms in the defining ideals of a fat point scheme and its support (see Conjectures 1.5 and 1.6) and they use a mixture of tools from Algebraic Geometry and Discrete Mathematics to study these relationships on initial degrees.NEWLINENEWLINESections 1 and 2 are devoted to give clear statements of the conjectures and the necessary background for symbolic powers of ideals of fat points and their Hilbert functions. Then, they prove Conjectures 1.5 and 1.6 for line count configurations \(W \subset {\mathbb P}^2\) of type \(c = (1, 2,\dots, t)\).NEWLINENEWLINEIn Section 3 the authors prove the conjectures for a more general case, i.e., for line count configurations of type \(c = (c_1,\dots, c_t)\) where \(c_i \geq i\) for each \(\leq i\leq t\).NEWLINENEWLINEIn Section 4, they give some interesting future directions.