A remark on a conjecture of Paranjape and Ramanan. We show that the spaces of global sections of exterior powers of a globally generated line bundle on a curve are not necessarily spanned by locally decomposable sections. The examples are based on the study of generic syzygy varieties. An application of these varieties is a short proof of Mukai’s theorem that every smooth curve of genus 7 and Clifford index 3 arises as the intersection of the spinor variety $Ssubset bfP^{15}$ with a transversal $bfP^6$.