Skip to main content Accessibility help
×
Home
Hostname: page-component-564cf476b6-2fphr Total loading time: 0.173 Render date: 2021-06-19T12:45:13.409Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

A specialization inequality for tropical complexes

Published online by Cambridge University Press:  30 April 2021

Dustin Cartwright
Affiliation:
Department of Mathematics, University of Tennessee, 227 Ayres Hall, Knoxville, TN 37996, USA cartwright@utk.edu
Corresponding
E-mail address:

Abstract

We prove a specialization inequality relating the dimension of the complete linear series on a variety to the tropical complex of a regular semistable degeneration. Our result extends Baker's specialization inequality to arbitrary dimension.

MSC classification

Type
Research Article
Copyright
© The Author(s) 2021

Access options

Get access to the full version of this content by using one of the access options below.

Footnotes

The author was supported by the National Science Foundation award number DMS-1103856 and National Security Agency Young Investigator Grant H98230-16-1-0019.

References

Baker, M., Specialization of linear systems from curves to graphs, Algebra Number Theory 2 (2008), 613653.10.2140/ant.2008.2.613CrossRefGoogle Scholar
Berkovich, V. G., Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33 (American Mathematical Society, Providence, RI, 1990).Google Scholar
Berkovich, V. G., Smooth $p$-adic analytic spaces are locally contractible, Invent. Math. 137 (1999), 184.10.1007/s002220050323CrossRefGoogle Scholar
Baker, M. and Norine, S., Riemann-Roch and Abel-Jacobi theory on a finite graph, Adv. Math. 215 (2007), 766788.CrossRefGoogle Scholar
Cartwright, D., Combinatorial tropical surfaces, Preprint (2015), arXiv:1506.02023.Google Scholar
Cartwright, D., Tropical complexes, Manuscripta Math. 163 (2020), 227261.CrossRefGoogle Scholar
Gathmann, A. and Kerber, M., A Riemann-Roch theorem in tropical geometry, Math. Z. 259 (2008), 217230.CrossRefGoogle Scholar
Hatcher, A., Algebraic topology (Cambridge University Press, Cambridge, 2002).Google Scholar
Jouanolou, J.-P., Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42 (Birkhäuser, Boston, 1983).Google Scholar
Kempf, G., Knudsen, F. F., Mumford, D. and Saint-Donat, B., Toroidal embeddings I, Lecture Notes in Mathematics, vol. 339 (Springer, Berlin, 1973).CrossRefGoogle Scholar
Lazarsfeld, R., Positivity in algebraic geometry I (Springer, Berlin, 2004).Google Scholar
Luo, Y., Rank-determining sets of metric graphs, J. Combin. Theory Ser. A 118 (2011), 17751793.CrossRefGoogle Scholar
Mikhalkin, G. and Zharkov, I., Tropical curves, their Jacobians and theta functions, in Curves and Abelian varieties, Contemporary Mathematics, vol. 465 (American Mathematical Society, Providence, RI, 2008), 203230.CrossRefGoogle Scholar
Nicaise, J., Singular cohomology of the analytic Milnor fiber, and mixed Hodge structure on the nearby cohomology, J. Algebraic Geom. 20 (2011), 199237.CrossRefGoogle Scholar
Ruggiero, M. and Shaw, K., Tropical Hopf manifolds and contracting germs, Manuscripta Math. 152 (2016), 160.CrossRefGoogle Scholar
Sumi, K., Tropical theta functions and Riemann–Roch inequality for tropical Abelian surfaces, Math. Z. 297 (2021), 13291351.CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A specialization inequality for tropical complexes
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

A specialization inequality for tropical complexes
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

A specialization inequality for tropical complexes
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *