##
**Tropical hyperelliptic curves.**
*(English)*
Zbl 1266.14050

Tropical geometry is a fairly recent trend in mathematics. It can be viewed as a degeneration of algebraic geometry. Therefore, there are fascinating connections between algebraic and tropical varieties, some of them yet to be discovered.

This paper brings together the tropical Brill–Noether theory (due to Baker, Caporaso, Cools, Draisma, Payne, Robeva, Lim, Potashnik, etc., since 2008) and moduli spaces of tropical curves (due to Mikhalkin, Zharkov, Caporaso, since 2006).

In this paper the author studies the locus of tropical hyperelliptic curves as a subset of the moduli space of tropical curves, for a fixed genus \(g\). A more general problem is to characterize the tropical Brill–Noether loci (of tropical curves of genus \(g\) admitting a divisor of degree \(d\) and rank at least \(r\), when \(g-(r+1)(g-d+r)<0\)), the hyperelliptic case being \(d=2\), \(r=1\).

M. Baker and S. Norine studied combinatorial graphs since 2007 [Adv. Math. 215, No. 2, 766–788 (2007; Zbl 1124.05049); Int. Math. Res. Not. 2009, No. 15, 2914–2955 (2009; Zbl 1178.05031)]. This notion evolved into that of metric graph, in close relation with the concept of abstract tropical curve. The author gives a new definition of harmonic morphism between metric graphs and proves that a metric graph \(\Gamma\) is a tropical hyperelliptic curve if and only if there exists a degree 2 harmonic morphism from \(\Gamma\) to a metric tree.

It is well–known that, for genus \(g\), the classical hyperelliptic locus is \((2g-1)\)-dimensional. Therefore, it is surprising that, in the tropical setting, it is \((3g-3)\)–dimensional. In this paper, the author proves that the locus of 2–edge–connected genus \(g\) tropical hyperelliptic curves is a \((2g-1)\)-dimensional stacky polyhedral fan whose maximal cells are in bijection with trees on \(g-1\) vertices and maximum valence 3.

At the end of the paper, the author concentrates on the tropicalization of hyperelliptic curves \(X\) in the plane, over a complete non–archimedean valuated field \(K\). If the curve \(X\) is given by a degree 2 polynomial whose Newton polygon has vertices \((0,0), (2g+2,0)\) and \((0,2)\), the author proves that under certain combinatorial conditions, the Berkovich skeleton of \(X\) is a ladder over the path on \(g-1\) vertices.

The paper is carefully written and very informative.

This paper brings together the tropical Brill–Noether theory (due to Baker, Caporaso, Cools, Draisma, Payne, Robeva, Lim, Potashnik, etc., since 2008) and moduli spaces of tropical curves (due to Mikhalkin, Zharkov, Caporaso, since 2006).

In this paper the author studies the locus of tropical hyperelliptic curves as a subset of the moduli space of tropical curves, for a fixed genus \(g\). A more general problem is to characterize the tropical Brill–Noether loci (of tropical curves of genus \(g\) admitting a divisor of degree \(d\) and rank at least \(r\), when \(g-(r+1)(g-d+r)<0\)), the hyperelliptic case being \(d=2\), \(r=1\).

M. Baker and S. Norine studied combinatorial graphs since 2007 [Adv. Math. 215, No. 2, 766–788 (2007; Zbl 1124.05049); Int. Math. Res. Not. 2009, No. 15, 2914–2955 (2009; Zbl 1178.05031)]. This notion evolved into that of metric graph, in close relation with the concept of abstract tropical curve. The author gives a new definition of harmonic morphism between metric graphs and proves that a metric graph \(\Gamma\) is a tropical hyperelliptic curve if and only if there exists a degree 2 harmonic morphism from \(\Gamma\) to a metric tree.

It is well–known that, for genus \(g\), the classical hyperelliptic locus is \((2g-1)\)-dimensional. Therefore, it is surprising that, in the tropical setting, it is \((3g-3)\)–dimensional. In this paper, the author proves that the locus of 2–edge–connected genus \(g\) tropical hyperelliptic curves is a \((2g-1)\)-dimensional stacky polyhedral fan whose maximal cells are in bijection with trees on \(g-1\) vertices and maximum valence 3.

At the end of the paper, the author concentrates on the tropicalization of hyperelliptic curves \(X\) in the plane, over a complete non–archimedean valuated field \(K\). If the curve \(X\) is given by a degree 2 polynomial whose Newton polygon has vertices \((0,0), (2g+2,0)\) and \((0,2)\), the author proves that under certain combinatorial conditions, the Berkovich skeleton of \(X\) is a ladder over the path on \(g-1\) vertices.

The paper is carefully written and very informative.

Reviewer: María Jesús de la Puente (Madrid)

### MSC:

14T05 | Tropical geometry (MSC2010) |

14H10 | Families, moduli of curves (algebraic) |

14H51 | Special divisors on curves (gonality, Brill-Noether theory) |

### Software:

OEIS### References:

[1] | Amini, O., Caporaso, L.: Riemann-Roch theory for weighted graphs and tropical curves. arXiv:1112.5134 · Zbl 1284.14087 |

[2] | Baker, M., Specialization of linear systems from curves to graphs, Algebra Number Theory, 2, 613-653, (2008) · Zbl 1162.14018 |

[3] | Baker, M.; Faber, X., Metric properties of the tropical Abel-Jacobi map, J. Algebr. Comb., 33, 349-381, (2011) · Zbl 1215.14060 |

[4] | Baker, M.; Norine, S., Riemann-Roch and Abel-Jacobi theory on a finite graph, Adv. Math., 215, 766-788, (2007) · Zbl 1124.05049 |

[5] | Baker, M.; Norine, S., Harmonic morphisms and hyperelliptic graphs, Int. Math. Res. Not., 15, 2914-2955, (2009) · Zbl 1178.05031 |

[6] | Baker, M., Payne, S., Rabinoff, J.: Nonarchimedean geometry, tropicalization, and metrics on curves. arXiv:1104.0320 · Zbl 1470.14124 |

[7] | Bieri, R.; Groves, J. R.J., The geometry of the set of characters induced by valuations, J. Reine Angew. Math., 347, 168-195, (1984) · Zbl 0526.13003 |

[8] | Brannetti, S.; Melo, M.; Viviani, F., On the tropical Torelli map, Adv. Math., 226, 2546-2586, (2011) · Zbl 1218.14056 |

[9] | Caporaso, L., Geometry of tropical moduli spaces and linkage of graphs, J. Comb. Theory, Ser. A, 119, 579-598, (2012) · Zbl 1234.14043 |

[10] | Caporaso, L.; Farkas, G. (ed.); Morrison, I. (ed.), Algebraic and tropical curves: comparing their moduli spaces, (2011) |

[11] | Caporaso, L.; Alexeev, V. (ed.); Izadi, E. (ed.); Gibney, A. (ed.); Kollár, J. (ed.); Loojenga, E. (ed.), Algebraic and combinatorial Brill-Noether theory, No. 564, (2012), Providence |

[12] | Cayley, A., On the analytical forms called trees, with application to the theory of chemical combinations, Rep. Br. Assoc. Adv. Sci., 45, 257-305, (1875) |

[13] | Chan, M.: Combinatorics of the tropical Torelli map. Algebra Number Theory (to appear). arXiv:1012.4539v2 |

[14] | Cools, F.; Draisma, J.; Payne, S.; Robeva, E., A tropical proof of the Brill-Noether Theorem, Adv. Math., 230, 759-776, (2012) · Zbl 1325.14080 |

[15] | Gathmann, A.; Kerber, M., A Riemann-Roch theorem in tropical geometry, Math. Z., 259, 217-230, (2008) · Zbl 1187.14066 |

[16] | Griffiths, P.; Harris, J., On the variety of special linear systems on a general algebraic curve, Duke Math. J., 47, 233-272, (1980) · Zbl 0446.14011 |

[17] | Haase, C.; Musiker, G.; Yu, J., Linear systems on tropical curves, Math. Z., 270, 1111-1140, (2012) · Zbl 1408.14201 |

[18] | Lim, C. M.; Payne, S.; Potashnik, N., A note on Brill-Noether theory and rank determining sets for metric graphs, Int. Math. Res. Not., (2012) · Zbl 1328.14099 |

[19] | Mikhalkin, G., Tropical geometry and its applications, No. II, 827-852, (2006), Zürich · Zbl 1103.14034 |

[20] | Mikhalkin, G.; Zharkov, I., Tropical curves, their Jacobians and theta functions, No. 465, 203-231, (2007) · Zbl 1152.14028 |

[21] | The On-Line Encyclopedia of Integer Sequences. Published electronically at http://oeis.org (2011) |

[22] | Rains, E. M.; Sloane, N. J.A., On Cayley’s enumeration of alkanes (or 4-valent trees), J. Integer Seq., 2, (1999) · Zbl 0999.05053 |

[23] | Urakawa, H., A discrete analogue of the harmonic morphism and Green kernel comparison theorems, Glasg. Math. J., 42, 319-334, (2000) · Zbl 1002.05049 |

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