Fat points on rational normal curves
The authors study the Hilbert function of schemes \(\mathbb{Z}\) of ``fat'' points in \(\mathbb{P}^3\), whose support lies on a rational normal curve \(C\) of degree \(r\). More precisely, let \(I_Z\) be a homogeneous ideal of type \({\mathfrak p}_1^{m_1}\cap \cdots\cap {\mathfrak p}^{m_s}_s\), where each \({\mathfrak p}_i\) is the homogeneous ideal in \(R=k[x_0, \dots,x_r]\) of a point \(P_i\) and \(m_1,\dots,m_s\) are integers. In this paper the authors consider the following conjectures: Conjecture A. The value of the Hilbert function \(H(R/I_Z,t)\) does not depend on the choice of the points on \(C\); Conjecture B. If \(W\) is a scheme of fat points supported on a set of points in linear general position, then \(H(R/I_Z,t)\leq H(R/I_W,t)\). Conjecture A was proved in the case \(r=2,3\) by \textit{M. V. Catalisano} and \textit{A. Gimigliano}, who gave an implicit algorithm (depending only on the numbers \(m_1,\dots,m_s)\) for computing \(H(R/I_Z,t)\) [see in particular J. Algebra 183, No. 1, 245-265 (1996; Zbl 0863.14028)]. Here, for the case \(g\geq 4\), the authors conjecture an algorithm (depending only on the \(m_i\)'s) to determine inductively \(H(R/I_Z,t)\). They also study the Hilbert function of the infinitesimal neighborhoods of a rational normal curve and they find the value where it coincides with the Hilbert polynomial. -- The main geometrical idea is to consider the base locus of the linear system \((I_Z)_t\). The authors prove that their algorithm works in several cases (and hence conjecture A in those cases): Either for \(s<r+2\), or for \(s\geq r+3\) and \(t\geq m_1+m_{r+3}-1\) (assuming \(m_1\geq \cdots\geq m_s)\); moreover, in case \(m_1=\cdots =m_s=m\) and either for \(s\leq r+2\) or for \(s>(m-1)(r-2)\); this implies conjecture A for \(m=2\) and any number of points [see also \textit{K. A. Chandler} in: Zero-dimensional schemes, Conf. Ravello 1992, 65-79 (1994; Zbl 0830.14019)]. Conjecture B is proved in the case \(r=2\).
infinitesimal neighborhoods of a rational normal curve
scientific article; zbMATH DE number 1340592