Enriched curves and their tropical counterpart
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, p. 689-741
In her Ph.D. thesis, Mainò introduced the notion of enriched structure on stable curves and constructed their moduli space. In this paper we give a tropical notion of enriched structure on tropical curves and construct a moduli space parametrizing these objects. Moreover, we use this construction to give a toric description of the scheme parametrizing enriched structures on a fixed stable curve.
Dans sa thèse, Mainò a introduit la notion de structure enrichie sur les courbes stables et elle a construit leur espace des modules. Dans cet article, nous donnons une notion tropicale de structure enrichie sur les courbes tropicales et nous construisons un espace de modules qui paramètre ces objets. De plus, nous utilisons cette construction pour donner une description torique du schéma qui paramètre les structures enrichies sur une courbe stable fixée.
Received : 2015-01-28
Revised : 2016-07-26
Accepted : 2016-09-15
Published online : 2017-05-31
DOI : https://doi.org/10.5802/aif.3095
Classification:  14H10,  14T05
Keywords: Enriched curve, tropical curve, moduli space
@article{AIF_2017__67_2_689_0,
     author = {Abreu, Alex C. and Pacini, Marco},
     title = {Enriched curves and their tropical counterpart},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {2},
     year = {2017},
     pages = {689-741},
     doi = {10.5802/aif.3095},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_2_689_0}
}
Abreu, Alex C.; Pacini, Marco. Enriched curves and their tropical counterpart. Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 689-741. doi : 10.5802/aif.3095. https://aif.centre-mersenne.org/item/AIF_2017__67_2_689_0/

[1] Abramovich, Dan; Caporaso, Lucia; Payne, Sam The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér., Tome 48 (2015) no. 4, pp. 765-809 | Article

[2] Amini, Omid; Caporaso, Lucia Riemann–Roch theory for weighted graphs and tropical curves, Adv. Math., Tome 240 (2013), pp. 1-23 | Article

[3] Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip A. Geometry of algebraic curves, Volume II, Springer, Grundlehren der Mathematischen Wissenschaften, Tome 268 (2011), xxii+928 pages (With a contribution by J. Harris.)

[4] Baker, Matthew; Norine, Serguei Riemann–Roch and Abel–Jacobi theory on a finite graph, Adv. Math., Tome 215 (2007) no. 2, pp. 766-788 | Article

[5] Batyrev, Victor; Blume, Mark The functor of toric varieties associated with Weyl chambers and Losev–Manin moduli spaces, Tohoku Math. J., Tome 63 (2011) no. 4, pp. 581-604 | Article

[6] Brannetti, Silvia; Melo, Margarida; Viviani, Filippo On the tropical Torelli Map, Adv. Math., Tome 226 (2011) no. 3, pp. 2546-2586 | Article

[7] Caporaso, Lucia Algebraic and tropical curves: comparing their moduli spaces, Handbook of Moduli, Volume I, International Press (Advanced Lectures in Mathematics) Tome 24 (2012), pp. 119-160

[8] Caporaso, Lucia Geometry of tropical moduli spaces and linkage of graphs, J. Comb. Theory, Ser., Tome 119 (2012) no. 3, pp. 579-598 | Article

[9] Caporaso, Lucia; Viviani, Filippo Torelli theorem for graphs and tropical curves, Duke Math. J., Tome 153 (2010), pp. 129-171 | Article

[10] Cavalieri, Renzo; Hampe, Simon; Markwig, Hannah; Ranganathan, Dhruv Moduli spaces of rational weighed stable curves and tropical geometry (2016) (https://arxiv.org/abs/1404.7426, to appear in Forum of Mathematics)

[11] Cavalieri, Renzo; Markwig, Hannah; Ranganathan, Dhruv Tropicalizing the space of admissible covers, Math. Ann., Tome 364 (2016) no. 3-4, pp. 1275-1313 | Article

[12] Cools, Filip; Draisma, Jan; Payne, Sam; Robeva, Elina A tropical proof of the Brill–Noether Theorem, Adv. Math., Tome 230 (2012) no. 2, pp. 759-776 | Article

[13] Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford Introduction to algorithms, MIT Press (2009), xix+1292 pages

[14] Cox, David A.; Little, John B.; Schenck, Henry K. Toric varieties, American Mathematical Society, Graduate Studies in Mathematics, Tome 124 (2011), xxiv+841 pages

[15] Eisenbud, David; Harris, Joe Limit linear series: Basic theory, Invent. Math., Tome 85 (1986), pp. 337-371 | Article

[16] Esteves, Eduardo Linear systems and ramification points on reducible nodal curves, Mat. Contemp., Tome 14 (1998), pp. 21-35

[17] Esteves, Eduardo; Medeiros, Nivaldo Limit canonical systems on curves with two components, Invent. Math., Tome 149 (2002) no. 2, pp. 267-338 | Article

[18] Hassett, Brendan Moduli spaces of weighted pointed stable curves, Adv. Math., Tome 173 (2003) no. 2, pp. 316-352 | Article

[19] Kapranov, Mikhail Veronese curves and Grothendieck-Knudsen moduli space M ¯ 0,n , J. Algebr. Geom., Tome 2 (1993) no. 2, pp. 239-262

[20] Li, Binglin Images of Rational Maps of Projective Spaces (2014) (https://arxiv.org/abs/1310.8453 )

[21] Mainò, Laila Moduli space of enriched stable curves, Harvard University (UK) (1998) (Ph. D. Thesis)

[22] Mikhalkin, Grigory Moduli spaces of rational tropical curves, Proceedings of Gökova Geometry-Topology Conference 2006, International Press (2007), pp. 39-51

[23] Mikhalkin, Grigory; Zharkov, Ilia Tropical curves, their Jacobians and Theta functions, Proceedings of the International Conference on Curves and Abelian Varieties in Honor of Roy Smith’s 65th Birthday, American Mathematical Society (Contemporary Mathematics) Tome 465 (2007), pp. 203-231

[24] Osserman, Brian A limit linear series moduli scheme, Ann. Inst. Fourier, Tome 56 (2006) no. 6, pp. 1165-1205 | Article

[25] Osserman, Brian Limit linear series for curves not of compact type (2014) (https://arxiv.org/abs/1406.6699 )

[26] Rizzo, Pedro Level-δ and stable limit linear series on singular curves, Instituto Nacional de Matemática Pura e Aplicada (Brazil) (2013) (Ph. D. Thesis)

[27] Ulirsch, Martin Tropical geometry of moduli spaces of weighted stable curves, J. Lond. Math. Soc., Tome 92 (2015) no. 2, pp. 427-450 | Article