2006/05/15 by Brian Osserman, Osserman, Brian
Mathematics · Physics and Astronomy · #14H51 #14M15 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Nonlinear Waves and Solitons #math.AG #msc:14H51 #msc:14M15
paper · pdf · doi:10.48550/arxiv.math/0605372
19 pages
arxiv created 2006/05/15 · openalex publication_date 2006/05/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In math.AG/0407496, a new construction of limit linear series is presented which functorializes and compactifies the original construction of Eisenbud and Harris, using a new space called the linked Grassmannian. The boundary of the compactification consists of crude limit series, and maps with positive-dimensional fibers to crude limit series of Eisenbud and Harris. In this paper, we carry out a careful analysis of the linked Grassmannian to obtain an upper bound on the dimension of the fibers of the map on crude limit series, thereby concluding an upper bound on the dimension of the locus of crude limit series, and obtaining a simple proof of the Brill-Noether theorem using only the limit linear series machinery. We also see that on a general reducible curve, even crude limit series may be smoothed to nearby fibers.