| Popis: |
We classify rank 2 vector bundles on a smooth curve X of genus 1 over a discrete valuation ring R. Atiyah [5] classified rank 2 vector bundles on elliptic curves over algebraically closed fields. The fact that a genus 1 curve over a discrete valuation ring has a codimension 2 subscheme prevents us from applying Atiyah's work directly. We find that genus 1 curve over an arbitrary field can have three types of rank 2 vector bundles. We classify rank 2 vector bundles on a curve of genus 1 over a discrete valuation ring using the classification on a curve of genus 1 over a field and quadruples (L,M,Z, η) where L and M are line bundles on X and Z is a local complete intersection subscheme of codimension 2 and η is an orbit in Ext¹(M⊗I(z),L ) under the R* action. |
