The union of two linear star configurations in \(\mathbb{P}^2\) all have generic Hilbert function
Let \(\{ F_i \}, 1 \leq i \leq t\) be general forms of respective degrees \(d_1,\dots,d_t\) in \(k[x_0,\dots,x_n]\), where \(k\) is a field. The author calls the codimension \(r \geq 2\) variety \(X\) defined by the ideal \(\bigcap_{1 \leq i_1 < \cdots < i_r \leq s} (F_{i_1}, \dots, F_{i_r})\) a star configuration in \(\mathbb{P}^n\), and if the \(H_i\) are all linear then he calls \(X\) a linear star configuration. (In the literature there is some variation on different aspects of these definitions.) When \(n=2\) we call \(t\) the type of the star configuration. In this paper the author resolves a question he had posed in an earlier paper (2012). Specifically, if \(X\) and \(Y\) are linear star configurations in \(\mathbb{P}^2\) of types \(s\) and \(t\) respectively (note that they are chosen generally by definition), then he asked if they have generic Hilbert function. That is, do they impose the maximum possible number of conditions on forms of every degree? The author gives an affirmative answer to this question. To achieve this, he specializes to the case of the union of a linear star configuration of type \(s\) and a fat point scheme of multiplicity \((t-1)\) in \(\mathbb{P}^2\), proves this case, and then applies it to the case of two linear star configurations. He gives an application to the case of codimension 2 linear star configurations in \(\mathbb{P}^3\). That is, if \(X\) and \(Y\) are such configurations with \(s,t \geq 3\), the author shows that the artinian algebra \(R/(I_X + I_Y)\) has the Weak Lefschetz Property.
scientific article; zbMATH DE number 7628606
scientific article