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Line arrangements and configurations of points with an unexpected geometric property. (English) Zbl 1408.14174

Let \(V = [R]_{j}\) be the vector space of degree \(j\) forms in three variables over an infinite field \(K\) and let \(\mathcal{L}_{j}\) be its projectivization. Consider \(X = m_{1}P_{1} + \dots + m_{r}P_{r}\) a fat point scheme supported on a set of \(r\) points \(P_{1}, \dots, P_{r} \in \mathbb{P}^{2}_{K}\). Then \(X\) is defined as \[ I_{X} = I(P_{1})^{m_{1}} \cap \dots \cap I(P_{r})^{m_{r}}. \] We say that \(X\) fails to impose the expected number of conditions on \(V\) (or on \(\mathcal{L}_{j}\)) if \[ \dim_{K} [I_{X}]_{j} > \max \bigg\{ 0, j + \binom{2}{2} - \sum_i m_i+\binom{1}{2}\bigg\}. \] If the points \(P_{i}\) are general, it is a well-known and difficult open problem to classify all \(m_{i}\) and \(j\) such that the subscheme \(X\) fails to impose expected number of conditions on \(V\), but a conjectural answer is delivered by the celebrated SHGH conjecture.
Conjecture (SHGH). For \(X = m_{1}P_{1} + \dots + m_{r}P_{r}\) with general points \(P_{i}\), \(X\) fails to impose the expected number of conditions on \(V\) only if \([I_{X}]_{j} \neq 0\), but the base locus of \([I_{X}]_{j}\) contains a multiple of a rational curve of a prescribed kind.
The main goal of the paper under review is to provide a generalization of the SHGH conjecture, emphasizing a very special role of certain classes of line arrangements in projective planes in that picture. It is worth mentioning here an example provided by R. Di Gennaro et al. [J. Lond. Math. Soc., II. Ser. 89, No. 1, 194–212 (2014; Zbl 1290.13013)] since this is the leitmotif in the whole story. In the mentioned paper, the authors observed that the set of nine points in \(\mathbb{P}^{2}_{K}\) dual to the so-called \(B_{3}\) arrangement has an unusual geometric property, namely for every point \(P\) on the plane there is a curve of degree \(4\) passing through these nine points and vanishing to order \(3\) at \(P\). The punchline in this story is that such a curve is in general unexpected – a naive dimension count suggests that the linear system of curves of degree \(4\) containing the nine points and \(3P\) should be empty except for a special locus of points \(P\), but in fact it is non-empty for a general point \(P\). The above example leads to the following setting: one wants to study finite sets of points \(Z\) in the plane for which, for some integer \(j\), the dimension of the linear system of plane curves of degree \(j+1\) that pass through the points of \(Z\) and have multiplicity \(j\) at a general point \(P\) is unexpectedly large. In such a case, we say that the set \(Z\) admits an unexpected curve of degree \(j+1\). In order to formulate the main result of the paper, let us define three invariants. For a finite set of points \(Z \subset \mathbb{P}^{2}_{K}\) we define the multiplicity index, denoted by \(m_{Z}\), as the least integer \(j\) such that the linear system of degree \(j+1\) forms vanishing at \(Z + jP\) is non-empty. Another one, which is new, is \[ t_{Z} := \min \bigg\{ j \geq 0 : h^{0}(\mathcal{I}_{Z}(j+1) ) - { j+\binom{1}{2}} > 0 \bigg\}, \] and, as it was observed by the authors, it depends only on the Hilbert function of \(Z\). Finally, we define the speciality index \(u_{Z}\) to be the least integer \(j\) such that the scheme \(Z + jP\), where \(P\) is a general point, imposes independent conditions on forms of degree \(j+1\).
Theorem 1. \(Z\) admits an unexpected curve if and only if \(m_{Z} < t_{Z}\). Furthermore, in the case \(Z\) has an unexpected curve of degree \(j\) if and only if \(m_{Z} < j \leq u_{Z}\).
In particular, the existence of an unexpected curve forces that \(m_{Z} < u_{Z}\), and the converse is almost but not quite true. This observation leads to the following geometric version of Theorem 1.
Theorem 2. Let \(Z \subset \mathbb{P}^{2}\) be a finite set of points. Then \(Z\) admits an unexpected curve if and only if \(2m_{Z} + 2 < |Z|\), but no subset of \(m_{Z}+2\) (or more) of the points is collinear. In that case, \(Z\) has an unexpected curve of degree \(j\) if and only if \(m_{Z} < j \leq |Z| - m_{Z}-2\).
Several examples of unexpected curves are studied in detail in Section 6, and there a special role is played by line arrangements which are free (in Saito’s sense). In the last section of the paper, the most interesting from my very subjective point of view, the authors present interesting connections of unexpected curves with the Strong Lefschetz Propery, and they rephrase Terao’s conjecture on the freeness of line arrangements using the language of splitting types of vector bundles.

MSC:

14N20 Configurations and arrangements of linear subspaces
13D02 Syzygies, resolutions, complexes and commutative rings
14C20 Divisors, linear systems, invertible sheaves
14N05 Projective techniques in algebraic geometry
05E40 Combinatorial aspects of commutative algebra
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)

Citations:

Zbl 1290.13013
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References:

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