On the geography of line arrangements (Q2127865)

In the paper under review, the authors study the so-called (combinatorial) geography problem for line arrangements. Here we would like to emphasize that the authors rediscover, in a certain way, the geography problem of log surfaces of general type that are associated with line arrangements in the complex projective plane. However, here the authors look at the combinatorial side instead of digging into the world of sheaves of logarithmic differentials. Let \(\mathcal{A} \subset \mathbb{P}^{2}_{\Bbbk}\) be an arrangement of \(d\) lines in the plane defined over an arbitrary field \(\Bbbk\). We define by \(t_{m}(\mathcal{A}) :=t_{m}\) the number of \(m\)-fold points of \(\mathcal{A}\), i.e., points in the plane where exactly \(m\) lines meet. We define integers \[\overline{c}_{1}^{2}(\mathcal{A}) = 9-5d + \sum_{m\geq 2}(3m-4)t_{m},\] \[\overline{c}_{2}(\mathcal{A}) = 3-2d + \sum_{m\geq 2}(m-1)t_{m}.\] They are called the Chern numbers of \(\mathcal{A}\). It is known that if for an arrangement \(\mathcal{A}\) one has \(t_{d}=t_{d-1}=0\), then the Chern numbers of \(\mathcal{A}\) are positive. Analogously to the geography problem for surfaces of general type, we can talk about the geography problem for line arrangements over a fixed field \(\Bbbk\): given a pair of integers \((a,b) \in \mathbb{Z}^{2}\), is there a line arrangement over \(\Bbbk\) with \(\overline{c}_{1}^{2}(\mathcal{A}) = a\) and \(\overline{c}_{2}(\mathcal{A})=b\)? Using a technical density lemma (proved as Lemma 4.2 therein), the authors show that the set of accumulation points of Chern slopes of arrangement over \(\overline{\mathbb{F}}_{p}\) is the interval \([2,3]\), and they also (re)prove the fact that the set of accumulation points of Chern slopes of arrangement over \(\mathbb{R}\) is the interval \([2,5/2]\). Let us also recall the notion of the linear Harbourne index for a line arrangement \(\mathcal{A} \subset \mathbb{P}^{2}_{\Bbbk}\), namely \[H_{L}(\mathcal{A}) = \frac{d - \sum_{m\geq 2}mt_{m}}{\sum_{m \geq 2}t_{m}}.\] It can be observed that \[H_{L}(\mathcal{A}) = \frac{3-(\overline{c}_{1}^{2}(\mathcal{A}) - 2\overline{c}_{2}(\mathcal{A}))}{d-(\overline{c}_{1}^{2}(\mathcal{A}) - 3\overline{c}_{2}(\mathcal{A}))}-2.\] As a result, the authors observe that if \(r\) is an accumulation point of \(H_{L}\), then \(r \in [-2, -\infty[\) and for real line arrangements the set of accumulation points for \(H_{L}\) is \([-3,-2]\). At the end of the paper, the authors propose two interesting open conjectures. Conjecture 1. The set of accumulation points of Chern slopes of arrangements over \(\mathbb{C}\) is \([2,5/2]\). Conjecture 2. The set of accumulation points of \(H_{L}\) for line arrangements over \(\mathbb{C}\) is \([-3,-2]\). Conjecture 3. The set of accumulation points of Chern slopes of arrangements over \(\mathbb{Q}\) is \([2,5/2]\).