Linear systems on K3-sections
A ``K3-section'' is a smooth curve of genus at least two which lies on a K3 surface. The types of special linear systems which exist on such a curve often depend only on the linear equivalence class of the curve rather than on the curve itself. The first examples of this phenomenon where given by \textit{B. Saint-Donat}, who studied linear systems of type \(g^ 1_ 2\) and \(g^ 1_ 3\); this work was extended by \textit{M. Reid} [J. Lond. Math. Soc., II. Ser. 13, 454-458 (1976; Zbl 0339.14021)] to more general \(g^ 1_ d\)'s. The first part of this paper analyzes a natural construction (based on the geometric Riemann-Roch theorem) to study the circumstances under which this ``propagation'' of linear systems from one curve to another occurs. Some examples given by Reid and others given in this paper suggested to M. Green that the correct generalizations of Reid's results should involve the ``Clifford index'' \(\nu:=d-2r\) of a special linear system of type \(g^ r_ d\). Specifically, Green conjectured that the minimum value of this Clifford index should depend only on the linear equivalence class of the curve on the K3 surface. An extension of Green's conjecture is proposed in this paper to the effect that any special linear system \(| Z|\) on the curve should be contained in one whose Clifford index is not larger than that of \(| Z|\), and which is cut out by a linear system on the K3 surface. This extended conjecture is then proved for linear systems of type \(g^ 1_ d.\) Some related results (including a proof of Green's conjecture) have been obtained by \textit{M. Green} and \textit{R. Lazarsfeld} [Invent. Math. 89, 357-370 (1987; Zbl 0625.14022)].
scientific article; zbMATH DE number 4017060
scientific article