1999/11/10 by Eduardo Esteves, Esteves, Eduardo, Mathieu Gagné +5 · 2 citations
Mathematics · Medicine · Pharmacology, Toxicology and Pharmaceutics · #14H20 (Secondary) #14H40 (Primary) 14K30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #Berberine and alkaloids research #FOS: Mathematics #math.AG #msc:14H20 #msc:14H40 #msc:14K30
paper · pdf · doi:10.48550/arxiv.math/9911071
Plain TeX, 21 pages
arxiv created 1999/11/10 · openalex publication_date 1999/11/10 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the following autoduality theorem for an integral projective curve C in any characteristic. Given an invertible sheaf L of degree 1, form the corresponding Abel map AL: C->J, which maps C into its compactified Jacobian, and form its pullback map AL^*: Pic0J to J, which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, points of multiplicity 2, then AL^* is an isomorphism, and forming it commutes with specializing C. Much of our work is valid, more generally, for a family of curves with, at worst, points of embedding dimension 2. In this case, we use the determinant of cohomology to construct a right inverse to AL^*. Then we prove a scheme-theoretic version of the theorem of the cube, generalizing Mumford's, and use it to prove that AL^* is independent of the choice of L. Finally, we prove our autoduality theorem: we use the presentation scheme to achieve an induction on the difference between the arithmetic and geometric genera; here, we use a few special properties of points of multiplicity 2.