# A remark on a conjecture of Paranjape and Ramanan

• ### Friedrich Eusen

Düsseldorf, Germany
• ### Frank-Olaf Schreyer

Universität des Saarlandes, Saarbrücken, Germany

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## Abstract

In this note, 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 $S \subset \mathbb P^{15}$ with a transversal $\mathbb P^6$.