Tropical superelliptic curves
-
Madeline Brandt
and
Paul Alexander Helminck
Abstract
We present an algorithm for computing the Berkovich skeleton of a superelliptic curve yn = f(x) over a valued field. After defining superelliptic weighted metric graphs, we show that each one is realizable by an algebraic superelliptic curve when n is prime. Lastly, we study the locus of superelliptic weighted metric graphs inside the moduli space of tropical curves of genus g.
Funding source: National Science Foundation
Award Identifier / Grant number:
Funding statement: Madeline Brandt was supported by a National Science Foundation Graduate Research Fellowship.
Communicated by: R. Cavalieri
Acknowledgements
The authors would like to thank the Max Planck Institute for Mathematics in the Sciences for their hospitality while they carried out this project.
References
[1] O. Amini, M. Baker, E. Brugallé, J. Rabinoff, Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta. Res. Math. Sci. 2 (2015), Art. 7, 67 pages. MR3375652 Zbl 1327.1411710.1186/s40687-014-0019-0Search in Google Scholar
[2] M. F. Atiyah, I. G. Macdonald, Introduction to commutative algebra. Westview Press, Boulder, CO 2016. MR3525784 Zbl 1351.13002Search in Google Scholar
[3] M. Baker, Algebraic Number Theory Course Notes. Fall 2006.Search in Google Scholar
[4] M. Baker, Specialization of linear systems from curves to graphs. Algebra Number Theory2 (2008), 613–653. MR2448666 Zbl 1162.1401810.2140/ant.2008.2.613Search in Google Scholar
[5] M. Baker, X. Faber, Metrized graphs, electrical networks, and Fourier analysis. Preprint 2005, arXiv:math/0407428v2 [math.AG]Search in Google Scholar
[6] V. G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, volume 33 of Mathematical Surveys and Monographs. Amer. Math. Soc. 1990. MR1070709 Zbl 0715.14013Search in Google Scholar
[7] B. Bolognese, M. Brandt, L. Chua, From curves to tropical Jacobians and back. In: Combinatorial algebraic geometry, volume 80 of Fields Inst. Commun., 21–45, Fields Inst. Res. Math. Sci., Toronto, ON 2017. MR3752493 Zbl 1390.1419210.1007/978-1-4939-7486-3_2Search in Google Scholar
[8] S. Bosch, Lectures on formal and rigid geometry.Springer 2014. MR3309387 Zbl 1314.1400210.1007/978-3-319-04417-0Search in Google Scholar
[9] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), 235–265. MR1484478 Zbl 0898.6803910.1006/jsco.1996.0125Search in Google Scholar
[10] S. Brannetti, M. Melo, F. Viviani, On the tropical Torelli map. Adv. Math. 226 (2011), 2546–2586. MR2739784 Zbl 1218.1405610.1016/j.aim.2010.09.011Search in Google Scholar
[11] R. Cavalieri, H. Markwig, D. Ranganathan, Tropicalizing the space of admissible covers. Math. Ann. 364 (2016), 1275–1313. MR3466867 Zbl 1373.1406410.1007/s00208-015-1250-8Search in Google Scholar
[12] M. Chan, Tropical hyperelliptic curves. J. Algebraic Combin. 37 (2013), 331–359. MR3011346 Zbl 1266.1405010.1007/s10801-012-0369-xSearch in Google Scholar
[13] D. Eisenbud, Commutative algebra. Springer 1995. MR1322960 Zbl 0819.1300110.1007/978-1-4612-5350-1Search in Google Scholar
[14] A. Grothendieck, Revêtements étales et groupe fondamental. Springer 1971. MR0354651 Zbl 0234.1400210.1007/BFb0058656Search in Google Scholar
[15] S. Grushevsky, M. Moeller, Explicit formulas for infinitely many Shimura curves in genus 4. Asian J. Math. 22 (2018), 381–390. MR3824574 Zbl 1393.1402010.4310/AJM.2018.v22.n2.a12Search in Google Scholar
[16] P. A. Helminck, Tropical Igusa invariants and torsion embeddings. Preprint 2016, arXiv:1604.03987 [math.AG]Search in Google Scholar
[17] P. A. Helminck, Tropicalizing abelian covers of algebraic curves. Preprint 2018, arXiv:1703.03067v2 [math.AG]Search in Google Scholar
[18] Q. Liu, Courbes stables de genre$2$ et leur schéma de modules. Math. Ann. 295 (1993), 201–222. MR1202389 Zbl 0819.1401010.1007/BF01444884Search in Google Scholar
[19] Q. Liu, Algebraic geometry and arithmetic curves. Oxford Univ. Press 2002. MR1917232 Zbl 0996.1400510.1093/oso/9780198502845.001.0001Search in Google Scholar
[20] Q. Liu, D. Lorenzini, Models of curves and finite covers. Compositio Math. 118 (1999), 61–102. MR1705977 Zbl 0962.1402010.1023/A:1001141725199Search in Google Scholar
[21] D. Maclagan, B. Sturmfels, Introduction to tropical geometry, volume 161 of Graduate Studies in Mathematics. Amer. Math. Soc. 2015. MR3287221 Zbl 1321.1404810.1090/gsm/161Search in Google Scholar
[22] M. Möller, Shimura and Teichmüller curves. J. Mod. Dyn. 5 (2011), 1–32. MR2787595 Zbl 1221.1403310.3934/jmd.2011.5.1Search in Google Scholar
[23] M. B. Monagan, K. O. Geddes, K. M. Heal, G. Labahn, S. M. Vorkoetter, J. M. J. McCarron, P. DeMarco, Maple 10 Programming Guide. Maplesoft, Waterloo ON, Canada 2005.Search in Google Scholar
[24] J. Neukirch, Algebraic number theory. Springer 1999. MR1697859 Zbl 0956.1102110.1007/978-3-662-03983-0Search in Google Scholar
[25] L. Pachter, B. Sturmfels, editors, Algebraic statistics for computational biology. Cambridge Univ. Press 2005. MR2205865 Zbl 1108.6211810.1017/CBO9780511610684Search in Google Scholar
[26] Q. Ren, S. V. Sam, B. Sturmfels, Tropicalization of classical moduli spaces. Math. Comput. Sci. 8 (2014), 119–145. MR3224624 Zbl 1305.1403110.1007/s11786-014-0185-xSearch in Google Scholar
[27] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves. Springer 1994. MR1312368 Zbl 0911.1401510.1007/978-1-4612-0851-8Search in Google Scholar
[28] The Stack Project Authors, Stacks Project. 2017. http://stacks.math.columbia.eduSearch in Google Scholar
[29] H. Stichtenoth, Algebraic function fields and codes. Springer 2009. MR2464941 Zbl 1155.1402210.1007/978-3-540-76878-4Search in Google Scholar
[30] M. Temkin, Moduli spaces of n-pointed stable curves. Lecture notes. www.math.huji.ac.il/∼temkin/teach/math647/lecture_notes.pdfSearch in Google Scholar
[31] R. J. Walker, Algebraic curves. Springer 1978. MR513824 Zbl 0399.1401610.1007/978-1-4612-6323-4Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Topology of tropical moduli of weighted stable curves
- Classification of slant surfaces in 𝕊3 × ℝ
- New dense superball packings in three dimensions
- Explicit computation of some families of Hurwitz numbers, II
- An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property
- Moduli of stable sheaves supported on curves of genus three contained in a quadric surface
- Exceptional points for finitely generated Fuchsian groups of the first kind
- Tropical superelliptic curves
- Differentiability of projective transformations in dimension 2
- On pseudo-Einstein real hypersurfaces
- On the moduli spaces of singular principal bundles on stable curves
- Three-dimensional connected groups of automorphisms of toroidal circle planes
Articles in the same Issue
- Frontmatter
- Topology of tropical moduli of weighted stable curves
- Classification of slant surfaces in 𝕊3 × ℝ
- New dense superball packings in three dimensions
- Explicit computation of some families of Hurwitz numbers, II
- An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property
- Moduli of stable sheaves supported on curves of genus three contained in a quadric surface
- Exceptional points for finitely generated Fuchsian groups of the first kind
- Tropical superelliptic curves
- Differentiability of projective transformations in dimension 2
- On pseudo-Einstein real hypersurfaces
- On the moduli spaces of singular principal bundles on stable curves
- Three-dimensional connected groups of automorphisms of toroidal circle planes
