Overview
- An introduction to the techniques of Geometric Invariant Theory via a detailed analysis of the GIT problem for polarized curves
- An introduction to the problem of compactifying moduli spaces through an interpretation of the output of the GIT analysis
- An introduction to the rich theory of compactified Jacobians for singular curves via three explicit examples
- A detailed description of the quotient stacks associated to the different GIT quotients, illustrating the interplay between these two techniques
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2122)
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Table of contents (17 chapters)
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Front Matter
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Back Matter
Keywords
About this book
We investigate GIT quotients of polarized curves. More specifically, we study the GIT problem for the Hilbert and Chow schemes of curves of degree d and genus g in a projective space of dimension d-g, as d decreases with respect to g. We prove that the first three values of d at which the GIT quotients change are given by d=a(2g-2) where a=2, 3.5, 4. We show that, for a>4, L. Caporaso's results hold true for both Hilbert and Chow semistability. If 3.5<a<4, the Hilbert semistable locus coincides with the Chow semistable locus and it maps to the moduli stack of weakly-pseudo-stable curves. If 2<a<3.5, the Hilbert and Chow semistable loci coincide and they map to the moduli stack of pseudo-stable curves. We also analyze in detail the critical values a=3.5 and a=4, where the Hilbert semistable locus is strictly smaller than the Chow semistable locus. As an application, we obtain three compactications of the universal Jacobian over the moduli space of stable curves, weakly-pseudo-stable curves and pseudo-stable curves, respectively.
Authors and Affiliations
Bibliographic Information
Book Title: Geometric Invariant Theory for Polarized Curves
Authors: Gilberto Bini, Fabio Felici, Margarida Melo, Filippo Viviani
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-319-11337-1
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2014
Softcover ISBN: 978-3-319-11336-4Published: 19 November 2014
eBook ISBN: 978-3-319-11337-1Published: 07 November 2014
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: X, 211
Number of Illustrations: 17 b/w illustrations
Topics: Algebraic Geometry