On the local negativity of surfaces with numerically trivial canonical class
Let \(X\) be a smooth complex projective surface. One of the most challenging problems in the study of negative curves on \(X\) is the bounded negativity conjecture, which claims that there exists an integer \(b(X)\) such that \(C^2 \geq - b(X)\) for every reduced curve \(C\) on \(X\). The number \(b(X)\) can be arbitrarily large depending on \(X\), as one can easily see by taking the blow-up of \(\mathbb P^2\) at \(s>>0\) distinct points. To approach the problem more effectively, the authors look at the so-called local Harbourne constant, a variation of the Harbourne constant considered by \textit{T. Bauer} et al. [Int. Math. Res. Not. 2015, No. 19, 9456--9471 (2015; Zbl 1330.14007)]. Given a configuration \(C: = \sum_{i=1}^n C_i\), where \(C_i\) are \(n\) mutually transverse smooth irreducible curves on \(X\), giving rise to \(s\) singular points, consider the blow-up \(Y \to X\) at these points, and let \(\widetilde{C}=\sum_{i=1}^n \widetilde{C_i}\), where \(\widetilde{C_i}\) is the proper transform of \(C_i\) in \(Y\). The local Harbourne constant of \(C \subset X\) is the rational number \(h(C):= {\widetilde{C}^2}/{s}\). The authors focus on the case of configurations \(C\) of \(n\) smooth rational curves with \(s\) singular points on surfaces \(X\) having numerically trivial canonical bundle. In this setting they prove that \(h(C) \geq - 4 + \frac{1}{ s}\big(2n+t-3c_2(X)\big)\), where \(t\) is the number of double points among the \(s\) singular points of \(C\). Then they produce several interesting examples of configurations \(C\) of smooth rational curves on \(K3\) and Enriques surfaces, showing that the bound above is not far from the precise value of \(h(C)\) they compute explicitly.
scientific article; zbMATH DE number 6898066