Geometric Invariant Theory for Polarized Curves / by Gilberto Bini, Fabio Felici, Margarida Melo, Filippo Viviani.

Format:

Book

Author/Creator:

Bini, Gilberto., Author.

Felici, Fabio., Author.

Melo, Margarida., Author.

Viviani, Filippo., Author.

Series:

Lecture Notes in Mathematics, 0075-8434 ; 2122

Language:

English

Subjects (All):

Geometry, Algebraic.

Algebraic Geometry.

Local Subjects:

Algebraic Geometry.

Physical Description:

1 online resource (X, 211 p. 17 illus.)

Edition:

1st ed. 2014.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2014.

Language Note:

English

Summary:

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.

Contents:

Introduction

Singular Curves

Combinatorial Results

Preliminaries on GIT

Potential Pseudo-stability Theorem

Stabilizer Subgroups

Behavior at the Extremes of the Basic Inequality

A Criterion of Stability for Tails

Elliptic Tails and Tacnodes with a Line

A Strati_cation of the Semistable Locus

Semistable, Polystable and Stable Points (part I)

Stability of Elliptic Tails

Semistable, Polystable and Stable Points (part II)

Geometric Properties of the GIT Quotient

Extra Components of the GIT Quotient

Compacti_cations of the Universal Jacobian

Appendix: Positivity Properties of Balanced Line Bundles. .

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references and index.

Description based on publisher supplied metadata and other sources.

ISBN:

3-319-11337-2

OCLC:

895007525