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On the postulation of lines and a fat line. (English) Zbl 1403.14018

In the paper under review, the authors study an interesting postulation problem on configurations of general lines in \(\mathbb{P}^{3}\). Before we formulate the main result, let us recall the key objects of study. Assume that \(\mathbb{K}\) is an arbitrary field of characteristic zero. Let \(X \subset \mathbb{P}^{n}_{\mathbb{K}}\) be a closed subscheme, then its Hilbert function is defined as \[ \text{HF}_{X} : \mathbb{Z} \ni d \longrightarrow \text{dim}_{\mathbb{K}}[S(X)]_{d} \in \mathbb{Z}, \] where \(S(X)\) denotes the graded homogeneous coordinate ring of \(X\). It is well-known that the Hilbert function becomes eventually (i.e., for large \(d\)) a polynomial. We denote the Hilbert polynomial of \(X\) by \(\text{HP}_{X}\). We shall say that \(X\) has a bipolynomial Hilbert function if \[ \text{HF}_{X}(d) = \text{min} \{ \text{HP}_{\mathbb{P}^{n}_{\mathbb{K}}}, \text{HP}_{X}(d)\} \] for all \(d\geq 1\).
Main Result. Let \(m\geq 1\) be a fixed integer. Then for \(d \geq d_{0}(m) = 3\cdot {m + 1\choose 3}\) the Hilbert function of a subscheme \(X \subset \mathbb{P}^{3}_{\mathbb{K}}\) consisting of \(r\geq 0\) general lines and one line of multiplicity \(m\) (i.e., defined by the \(m\)-th power of the ideal of a line) satisfies that \[ \text{HF}_{X}(d) = \text{min} \{ \text{HP}_{\mathbb{P}^{3}_{\mathbb{K}}}, \text{HP}_{X}(d)\} \] for all \(d\geq 1\).
In other words, a general fat line and an arbitrary number of general lines with multiplicity \(1\) impose independent conditions on forms of degree \(d \geq d_{0}(m)\).

MSC:

14C20 Divisors, linear systems, invertible sheaves
14F17 Vanishing theorems in algebraic geometry
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14N05 Projective techniques in algebraic geometry
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References:

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