# Geometry and Arithmetic

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#### DOI: 10.4171/119-1/7

A remark on a conjecture of Paranjape and Ramanan

Friedrich Eusen[1] and Frank-Olaf Schreyer

(1) Düsseldorf, Germany

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 $\mathbb P^6$.

Keywords: Generic syzygy varieties, vector bundles, curves of genus 7, Green’s conjecture