Tropical covers of curves and their moduli spaces. (English) Zbl 1318.14060

Buchholz, Arne; Markwig, Hannah

Commun. Contemp. Math. 17, No. 1, Article ID 1350045, 27 p. (2015).

Let \(\mathcal{L}\) be the abstract curve that corresponds to a generic tropical line in the tropical projective plane, i.e. a curve with one vertex that one denotes by \(c\) and three ends adjacent to \(c\) called \(u\), \(v\) and \(w\). Let \(h: \Gamma \to \mathcal{L}\) be a cover of degree \(d\). The weights of the ends mapping to \(u\), \(v\) and \(w\) give rise to partitions \(\Delta_u\), \(\Delta_v\) and \(\Delta_w\) of \(d\), and the triple \(\Delta = (\Delta_u, \Delta_v, \Delta_w)\) is called the ramification profile of \(h\). Now the authors introduce \(\mathrm{M}_g^{\mathrm{trop}}(\mathcal{L},\Delta)\), the moduli space of tropical covers of \(\mathcal{L}\) of genus \(g\) with ramification profile \(\Delta\).
The tropical branch map on \(\mathrm{M}_g^{\mathrm{trop}}(\mathcal{L},\Delta)\) is \(\mathrm{br}^{\mathrm{trop}}: \mathrm{M}_g^{\mathrm{trop}}(\mathcal{L},\Delta) \to \mathcal{L}^r\), \((h: \Gamma \to \mathcal{L}) \mapsto (h(V_1),h(V_2),\dots,h(V_r))\), with \(r := \#\Delta + 2g - 2 - d\) the total number of labels.
The main theorems are:
Theorem 2.15. The moduli space \(\mathrm{M}_g^{\mathrm{trop}}(\mathcal{L},\Delta)\) is an abstract weighted polyhedral complex of pure dimension \(r\).
Theorem 3.3. The degree of \(\mathrm{br}^{\mathrm{trop}}\) is constant, called the tropical Hurwitz number \(H_d^{g,\mathrm{trop}}(\Delta)\).
Theorem 3.6. The tropical Hurwitz numbers \(H_d^{g,\mathrm{trop}}(\Delta)\) defined using tropical intersection theory equal their algebraic counterparts \(H_d^g(\Delta)\).

Reviewer: Timo Keller (Kaiserslautern)

MSC:

14T05	Tropical geometry (MSC2010)
14N35	Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
51M20	Polyhedra and polytopes; regular figures, division of spaces

Keywords:

tropical geometry; Hurwitz numbers; covers of curves

References:

[1]	DOI: 10.4171/RSMUP/125-10 · Zbl 1226.14066 · doi:10.4171/RSMUP/125-10
[2]	DOI: 10.1016/j.jcta.2011.11.011 · Zbl 1234.14043 · doi:10.1016/j.jcta.2011.11.011
[3]	DOI: 10.1007/s10801-009-0213-0 · Zbl 1218.14058 · doi:10.1007/s10801-009-0213-0
[4]	DOI: 10.1016/j.aim.2011.06.021 · Zbl 1231.14023 · doi:10.1016/j.aim.2011.06.021
[5]	DOI: 10.1023/A:1014347115536 · Zbl 1054.14033 · doi:10.1023/A:1014347115536
[6]	DOI: 10.1112/S0010437X08003837 · Zbl 1169.51021 · doi:10.1112/S0010437X08003837
[7]	Gathmann A., Electron. J. Combin. 19 pp P26– (2012)
[8]	DOI: 10.1007/978-1-4757-3849-0 · doi:10.1007/978-1-4757-3849-0
[9]	DOI: 10.1090/conm/580/11500 · Zbl 1317.14118 · doi:10.1090/conm/580/11500
[10]	DOI: 10.4171/CMH/166 · Zbl 1205.14071 · doi:10.4171/CMH/166
[11]	Li J., J. Differential Geom. 57 pp 509– (2001) · Zbl 1076.14540 · doi:10.4310/jdg/1090348132
[12]	Li J., J. Differential Geom. 60 pp 199– (2002) · Zbl 1063.14069 · doi:10.4310/jdg/1090351102
[13]	DOI: 10.1090/S0894-0347-05-00477-7 · Zbl 1092.14068 · doi:10.1090/S0894-0347-05-00477-7