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 IP^1-bundle. Finally, using Abel maps of bidegree (m,1), we characterize the curves having double points at worst
Keywords
Cite
@article{arxiv.math/9911069,
title = {Abel Maps and Presentation Schemes},
author = {E. Esteves and Mathieu Gagne and S. Kleiman},
journal= {arXiv preprint arXiv:math/9911069},
year = {2007}
}
Comments
Plain TeX, 32 pages. This is the final version, to appear in the Hartshorne issue of the Communications in Algebra. A number of small improvements to the original exposition were made here and there. In addition, Lemma (3.7) was corrected by reformulating it over an algebraically closed base field, and its proof revised accordingly. Since (3.7) was used in the proofs of Proposition (6.2) and Theorem (6.3), those proofs were revised too, (6.2) extensively and (6.3) minimally