Theta-characteristics on tropical curves. (English) Zbl 1510.14024
Tropical geometry is a modern branch of algebraic geometry, which implements algebro-geometric ideas on certain piecewise linear, distinctly combinatorial, tropical data structures. In this sense, the tropical analogue of an algebraic curve is a tropical curve, consisting of a metric graph and some non-negative integers on the vertices (called vertex genera). Perhaps surprisingly, in this setting there is still a well-working theory of divisors on tropical curves and linear equivalence of these objects.
In the present article, the authors study tropical theta-characteristics, which are simply divisors \(D\) on a tropical curve \(\Gamma\) such that \(2D\) is linearly equivalent to the canonical divisor of \(\Gamma\). This definition is the obvious tropical analogue for theta-characteristics known from algebraic geometry. As a first result, the authors prove that tropical theta-characteristics on \(\Gamma\) are in one-to-one-correspondence to cyclic subgraphs of \(\Gamma\) (Proposition 3.5). Moreover, in Theorem 3.10 they classify which tropical theta-characteristics are effective, thus generalizing earlier work by I. Zharkov [Contemp. Math. 527, 165–168 (2010; Zbl 1213.14120)].
The authors then continue to describe a moduli space \(T_g^{\mathrm{trop}}\) parametrizing pairs of tropical curves and a theta-characteristic. This space is constructed as a generalized cone complex using the (by now well-established) methods from [D. Abramovich et al., Ann. Sci. Éc. Norm. Supér. (4) 48, No. 4, 765–809 (2015; Zbl 1410.14049)]. One could be tempted to think of \(T_g^{\mathrm{trop}}\) as a moduli space of tropical spin curves, but in fact, previous work by the authors [L. Caporaso et al., Sel. Math., New Ser. 26, No. 1, Paper No. 16, 44 p. (2020; Zbl 1442.14195)] has shown that for a good notion of a tropical spin curve extra data should be added. Hence there is a forgetful map from the moduli space of tropical spin curves to \(T_g^{\mathrm{trop}}\).
Finally, in Theorem 5.4 the authors give the number of algebraic lifts for any given tropical theta-characteristic. This improves on [D. Jensen and Y. Len, Sel. Math., New Ser. 24, No. 2, 1391–1410 (2018; Zbl 1420.14144)] in the sense that here vertex genera are taken into account.
In the present article, the authors study tropical theta-characteristics, which are simply divisors \(D\) on a tropical curve \(\Gamma\) such that \(2D\) is linearly equivalent to the canonical divisor of \(\Gamma\). This definition is the obvious tropical analogue for theta-characteristics known from algebraic geometry. As a first result, the authors prove that tropical theta-characteristics on \(\Gamma\) are in one-to-one-correspondence to cyclic subgraphs of \(\Gamma\) (Proposition 3.5). Moreover, in Theorem 3.10 they classify which tropical theta-characteristics are effective, thus generalizing earlier work by I. Zharkov [Contemp. Math. 527, 165–168 (2010; Zbl 1213.14120)].
The authors then continue to describe a moduli space \(T_g^{\mathrm{trop}}\) parametrizing pairs of tropical curves and a theta-characteristic. This space is constructed as a generalized cone complex using the (by now well-established) methods from [D. Abramovich et al., Ann. Sci. Éc. Norm. Supér. (4) 48, No. 4, 765–809 (2015; Zbl 1410.14049)]. One could be tempted to think of \(T_g^{\mathrm{trop}}\) as a moduli space of tropical spin curves, but in fact, previous work by the authors [L. Caporaso et al., Sel. Math., New Ser. 26, No. 1, Paper No. 16, 44 p. (2020; Zbl 1442.14195)] has shown that for a good notion of a tropical spin curve extra data should be added. Hence there is a forgetful map from the moduli space of tropical spin curves to \(T_g^{\mathrm{trop}}\).
Finally, in Theorem 5.4 the authors give the number of algebraic lifts for any given tropical theta-characteristic. This improves on [D. Jensen and Y. Len, Sel. Math., New Ser. 24, No. 2, 1391–1410 (2018; Zbl 1420.14144)] in the sense that here vertex genera are taken into account.
Reviewer: Felix Röhrle (Frankfurt am Main)
MSC:
| 14H10 | Families, moduli of curves (algebraic) |
| 14H40 | Jacobians, Prym varieties |
| 14T20 | Geometric aspects of tropical varieties |
References:
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