Let (X, H) be a polarized K3 surface with \(\mathrm {Pic}(X) = \mathbb {Z}H\), and let \(C\in |H|\) be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when \(g\ge r^2\ge 4\), the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree \(d(\ge 5)\) smooth plane curve is \(d-4\), which is the same as the Clifford index of the curve.

令 ( X ,  H ) 是一个极化的 K3 表面，其中\(\mathrm {Pic}(X) = \mathbb {Z}H\)，并让\(C\in |H|\)是一个平滑的 genus 曲线克。我们举一个上限上一个半稳定向量丛的全球部分的尺寸Ç。这使我们能够计算具有高属的C的更高等级的 Clifford 指数。特别地，当\(g\ge r^2\ge 4\) 时，C的秩r Clifford 指数可以通过对X上的 Lazarsfeld-Mukai 丛的限制来计算，对应于曲线C上的线丛. 这是 Green 和 Lazarsfeld 将 K3 曲面上的曲线结果推广到更高阶向量丛的结果。我们也将同样的方法应用到投影平面上，并证明了一个度数\(d(\ge 5)\)光滑平面曲线的秩r Clifford 指数是\(d-4\)，这与 Clifford 相同曲线指数。