Curves with low Harbourne constants on Kummer and abelian surfaces
In the paper under review, the author studies the so-called Harbourne constants which measure the local negativity property. Let us recall that for a smooth complex projective surface \(X\) it is conjectured that there exists an integer \(b(X)\) such that for all reduced curves (not necessarily irreducible) one has \(C^2\geq -b(X)\). This conjecture is called the bounded negativity conjecture and it is widely open. In the paper under review, the author presents interesting curve configurations in abelian or Kummer surfaces having low Harbourne constants. In order to formulate the main results, let us recall the notion of mentioned Harbourne constants. Definition. Let \(X\) be a smooth surface and \(C \subset X\) a reduced curve. Let \(\mathcal{P} \subset X\) be a finite non-empty set of points and denote by \(\pi: \tilde{X} \rightarrow X\) be the blowing up along \(\mathcal{P}\). The number \[ H(X,\mathcal{P}) = \frac{ \tilde{C}^{2}}{\# \mathcal{P}}, \] where \(\tilde{C}\) is the strict transform of \(C\), is called the local Harbourne constant. Then the Harbourne constant of \(C\) is defined as \[ H(C) = \text{inf}_{\mathcal{P}} H(C,\mathcal{P}), \] where \(\mathcal{P}\) varies among all finite non-empty subsets of points in \(X\). The global Harbourne constant of \(X\) is defined by \[ H(X) = \text{inf}_{C} H(C), \] where the infimum is taken over all reduced curves \(C \subset X\). Theorem A. Let \(A\) be a simple complex abelian surface. There exists a sequence of curves \(\{R_{n}\}_{n \in \mathbb{N}}\) such that \(H(R_{n}) = - 4\). If \(A\) is the Jacobian of a smooth genus \(2\) curve, then \(R_{1}\) can be chosen either as the union of \(16\) specific smooth curves or as a single irreducible curve. It is worth emphasizing that curves \(R_{n}\) are constructed with use of the multiplication map \(n \in \mathbb{Z}\). Theorem B. Let \(X\) be a Jacobian-Kummer complex surface. For any \(n>1\), there are configurations \(C_{n}\) of genus \(g > 1\) curves such that \(H(C_{n}) = -4 \frac{n^{4}}{n^{4}-1}\).
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