Divisibilities among nodal curves (Q1720126)

This paper proves the following theorem. Theorem. Let \(S\) ba a smooth algebraic surface. Let \(R\subset \text{Num}(S)\) be a (negative-definite) root lattice generated by \((-2)\) curves. Denote by the primitive closure by \(R^{\prime}=(R\otimes{\mathbb{Q}}) \cap\text{Num}(S)\), and let \(D\in\mathbb R^{\prime}\setminus R\). If \(D^2=-2\) or \(-1\), then \(D\) is neither effective nor anti-effective. The proof rests on basic properties of root lattices and their duals, in particular, in relations to reflections. As a corollary, an application to \(K3\) surfaces is discussed. Corollary: Let \(S\) be a \(K3\) surface and \(R\subset\text{Num}(S)\) be a root lattice generated by nodal curves on \(S\). It primitive closure \(R^{\prime}\) contains no vectors outside \(R\) with square \(-2\): \(D\in R^{\prime}, \, D^2>-4\), then \(D\in R\). Theorem has also an application on possible configurations of nodal curves on Enriques surfaces, in particular, on divisibilitoes of nordal curves.