Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-12T19:48:54.125Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Madhusudan Manjunath
Affiliation:
Department of Mathematics, Indian Institute of Technology, Bombay email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1] Abramovich, D., Caporaso, L., and Payne, S., The tropicalization of the moduli space of curves. Ann. Sci. Ée. Norm. Supér. 48(2015), 765–809. http://dx.doi.Org/10.24033/asens.2258 Google Scholar
[2] Abramovich, D., Caporaso, L., and Payne, S., Equidistribution of Weierstrass points on curves over non-Archimedean fields. arxiv:1412.0926.Google Scholar
[3] Amini, O. and Baker, M., Linear series on metrized complexes of algebraic curves. Math. Ann. 362(2015), no. 1-2, 55–106. http://dx.doi.org/10.1007/s00208-014-1093-8 Google Scholar
[4] Amini, O., Baker, M., Brugallé, E., and Rabinoff, J., Lifting harmonic morphisms II: tropical curves and metrized complexes. Algebra Number Theory 9(2015), 267–315. http://dx.doi.Org/10.2140/ant.2015.9.267 Google Scholar
[5] Amini, O., Baker, M., Brugalle, E., and Rabinoff, J., Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta. Res. Math. Sci. 2(2015), Art. 7, 67.Google Scholar
[6] Baker, M., Payne, S., and Rabinoff, J., Nonarchimedean geometry, tropicalization, and metrics on curves. Algebr. Geom. 3(2016), 63–105. http://dx.doi.org/10.14231/AC-2016-004 Google Scholar
[7] Baker, M. and Rumely, R., Potential theory and dynamics on the Berkovich projective line. Mathematical Surveys and Monographs, 159. American Mathematical Society, Providence, RI, 2010. http://dx.doi.Org/10.1090/surv/159 Google Scholar
[8] Berkovich, V. G., Spectral theory and analytic geometry over non-Archimedean fields. Mathematical Surveys and Monographs, 33. American Mathematical Society, Providence, RI, 1990.Google Scholar
[9] Billera, L., Holmes, S., and Vogtmann, K., Geometry of the space of phylogenetic trees. Adv. in Appl. Math. 27(2001), no. 4, 733–767. http://dx.doi.Org/10.1006/aama.2001.0759 Google Scholar
[10] Cartwright, D., Lifting matroid divisors on tropical curves. Res. Math. Sci. 2(2015), no. 1, 1–24. http://dx.doi.Org/10.1186/s40687-015-0041-x Google Scholar
[11] Cartwright, D., Jensen, D., and Payne, S., Lifting divisors on a generic chain of loops. Canad. Math. Bull. 58(2015), 250–262, 2015. http://dx.doi.Org/10.4153/CMB-2O14-050-2 Google Scholar
[12] Castelnuovo, G., Numero delle involuzioni razionali gaicenti sopra una curva di dato genere, rendi. R. Accad. Lincei 4(1889), 130–133.Google Scholar
[13] Cavalieri, R., Markwig, H., and Ranganathan, D., Tropicalizing the space of admissible covers. Math. Ann. 364(2016), no. 3-4, 1275–1313. http://dx.doi.org/10.1007/s00208-015-1250-8 Google Scholar
[14] Cools, F., Draisma, J., Payne, S., and Robeva, E., A tropical proof of the Brill-Noether theorem. Adv. Math. 230(2012), no. 2, 759–776. http://dx.doi.Org/10.1016/j.aim.2O12.02.019 Google Scholar
[15] DeMarco, L. and McMullen, C., Trees and the dynamics of polynomials. Ann. Sci. Ec. Norm. Super. 41(2008), 337–382. http://dx.doi.org/10.24033/asens.2070 Google Scholar
[16] Eisenbud, D. and Harris, J., Limit linear series: basic theory. Invent. Math. 85(1986), no. 2, 337–371. http://dx.doi.Org/10.1007/BF01389094 Google Scholar
[17] Eisenbud, D. and Harris, J., The Kodaira dimension of the moduli space of curves of genu. ≧ 23. Invent. Math. 90(1987), no. 2, 359–387. http://dx.doi.org/10.1007/BF01388710 Google Scholar
[18] Eisenbud, D. and Harris, J., The monodromy of Weierstrass points. Invent. Math. 90(1987), no. 2, 333–341. http://dx.doi.Org/10.1007/BF01388708 Google Scholar
[19] Eisenbud, D. and Harris, J., Irreducibility of some families of linear series with Brill-Noether number — 1. Ann. Sci. Éc. Norm. Super. 22(1989), 33–53. http://dx.doi.Org/10.24033/asens.1574 Google Scholar
[20] Foster, T., Introduction to adic tropicalization. arxiv:1506.00726Google Scholar
[21] Gathmann, A. and Kerber, M., A Riemann-Roch theorem in tropical geometry. Math. Z. 259(2008), no. 1, 217–230. http://dx.doi.org/10.1007/s00209-007-0222-4 Google Scholar
[22] Gieseker, D., Stable curves and special divisors: Petri's conjecture. Invent. Math. 66(1982), no. 2, 251–275. http://dx.doi.org/10.1007/BF01389394 Google Scholar
[23] Griffiths, P. and Harris, J., On the variety of special linear systems on a general algebraic curve. Duke Math. J. 47(1980), no. 1, 233–272. http://dx.doi.org/10.1215/S0012-7094-80-04717-1 Google Scholar
[24] Harris, J. and Mumford, D., On the Kodaira dimension of the moduli space of curves. Invent. Math. 67(1982), 1, 23–86. http://dx.doi.org/10.1007/BF01393371 Google Scholar
[25] Harris, J. and Morrison, I., Moduli of curves. Graduate Texts in Mathematics, 187. Springer-Verlag, New York, 1998.Google Scholar
[26] Jensen, D. and Payne, S., Tropical independence I: Shapes of divisors and a proof of the Gieseker-Petri theorem. Algebra Number Theory 8(2014), 2043–2066. http://dx.doi.Org/10.2140/ant.2014.8.2043 Google Scholar
[27] Jensen, D. and Ranganathan, D., Brill-Noether theory for curves of a fixed gonality. arxiv:1 701.06579Google Scholar
[28] Luo, Y., Tropical convexity and canonical projections. arxiv:1304.7963Google Scholar
[29] Markwig, T., A field of generalised Puiseux series for tropical geometry. Rend. Semin. Mat. Univ. Politec. Torino 68(2010), 79–92.Google Scholar
[30] Osserman, B., Limit linear series. 2012. https://www.math.ucdavis.edu/∼osserman/classes/280-W13/llsbook.pdf. Google Scholar
[31] Osserman, B., Limit linear series for curves not of compact type. arxiv:1406.6699, 2014.Google Scholar
[32] Payne, S., Fibers of tropicalization. Math. Z. 262(2009), no. 2, 301–311. http://dx.doi.Org/10.1007/s00209-008-0374-x Google Scholar