Abel Maps and Presentation Schemes
Abstract
We sharpen the two main tools used to treat the compactified Jacobian of a singular curve: Abel maps and presentation schemes. First we prove a smoothness theorem for bigraded Abel maps. Second we study the two complementary filtrations provided by the images of certain Abel maps and certain presentation schemes. Third we study a lifting of the Abel map of bidegree (m,1) to the corresponding presentation scheme. Fourth we prove that, if a curve is blown up at a double point, then the corresponding presentation scheme is a IP1-bundle. Finally, using Abel maps of bidegree (m,1), we characterize the curves having double points at worst
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.