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Smoothing of Limit Linear Series of Rank One on Saturated Metrized Complexes of Algebraic Curves

Published online by Cambridge University Press:  20 November 2018

Ye Luo  and
Madhusudan Manjunath
Ye Luo
Affiliation:
School of Information Science and Engineering, Xiamen University email: luoye80@gmail.com
Madhusudan Manjunath
Affiliation:
Department of Mathematics, Indian Institute of Technology, Bombay email: madhu@math.iitb.ac.in
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Abstract

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We investigate the smoothing problem of limit linear series of rank one on an enrichment of the notions of nodal curves and metrized complexes called saturated metrized complexes. We give a finitely verifiable full criterion for smoothability of a limit linear series of rank one on saturated metrized complexes, characterize the space of all such smoothings, and extend the criterion to metrized complexes. As applications, we prove that all limit linear series of rank one are smoothable on saturated metrized complexes corresponding to curves of compact-type, and we prove an analogue for saturated metrized complexes of a theorem of Harris and Mumford on the characterization of nodal curves contained in a given gonality stratum. In addition, we give a full combinatorial criterion for smoothable limit linear series of rank one on saturated metrized complexes corresponding to nodal curves whose dual graphs are made of separate loops.

Keywords

14T05limit linear seriesmetrized complex
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 3 , 01 June 2018 , pp. 628 - 682
Copyright
Copyright © Canadian Mathematical Society 2018

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