On Nikulin's theorem on fixed components of linear systems on \(K3\) surfaces
Let \(X\) be a smooth K3 surface and let \(D\) be an effective divisor on \(X\). One can construct the `dual weighted graph' of \(D\) by taking vertices for irreducible components (weights correspond to multiplicities) and edges representing the intersection numbers. \textit{V. V. Nikulin} [cf. Algebraic geometry and analytic geometry, Proc. Conf., Tokyo 1990, ICM-90 Satell. Conf. Proc., 138-164 (1991; Zbl 0785.14021)] described the weighted graphs arising from the previous construction, starting with a divisor \(D\) which is general in its linear system; in this case, writing \(D = C + \Delta\), \(\Delta=\) fixed part, then \(C\) must be irreducible unless it belongs to a multiple of an elliptic pencil, so the graph of \(D\) is essentially determined by the components of \(\Delta\) and their intersections with \(C\). The author provides here a new, direct proof of Nikulin's results.
scientific article; zbMATH DE number 469057