Limits of Compactified Jacobians and D-Modules on Smooth Projective Curves
✍ Scribed by Rob C. Cannings; Martin P. Holland
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 276 KB
- Volume
- 135
- Category
- Article
- ISSN
- 0001-8708
No coin nor oath required. For personal study only.
✦ Synopsis
Let S be a smooth projective curve and D S its sheaf of differential operators. This paper classifies the rank one torsion-free D S -modules up to isomorphism. Such a module E has a degree which depends on the homological properties of E. Furthermore, the set of isomorphism classes with fixed degree d, say, bijects to
the set of points of the limit of compactified Jacobians of curves with injective normalization ?: S Ä X. Here, if F is a rank one torsion-free O X -module (of degree d+$(X )) representing the point of the limit corresponding to a torsion-free D S -module E then E$D OS , F the sheaf of differential operators from O S to F. As an application, we classify the domains Morita equivalent to D(P 1 ). We also study D(F), the ring of globally defined differential operators on a rank one torsion-free sheaf over a curve with injective normalization P 1 Ä X. It is shown that the finitedimensional simple D(F)-modules are precisely H i (X, F), when they are non-zero, and that the (non)vanishing of H i (X, F) determines the Morita equivalence class of D(F).