A remark about the rigidity of curves on \(K3\) surfaces
A classical result due to Bogomolov and Mumford states that on any projective \(K3\) surface there are rational curves and 1-dimensional families of elliptic curves. Moreover, \textit{Xi Chen} [J. Algebr. Geom. 8, 245--278 (1999; Zbl 0940.14024)] has shown that if \(S \subset \mathbb{P}^n\) is any \(K3\) surface of Picard number 1, then the linear system \(|\mathcal{O}_S(d)|\) contains a \((p(d)-\delta)\)-dimensional family of irreducible nodal curves whose arithmetic genus is \(p(d)=d^2(n-1)+1\) and which possess exactly \(\delta\) nodes, for \(\delta\geq 0\). In this paper, the author gives another proof of this result but with certain restrictions on the number of nodes \(\delta\).
scientific article; zbMATH DE number 5939723