Superabundant Curves and the Artin Fan

2017

J. London Math. Soc.

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ABSTRACT. We study the Berkovich analytification of the space of genus 0 logarithmic stable maps to a toric variety X and present applications to both algebraic and tropical geometry. On algebraic side, insights from tropical geometry give two new geometric descriptions of this space of maps - (1) as an explicit toroidal modification of M 0,n ×X and (2) as a tropical compactification in a toric variety. On the combinatorial side, we prove that the tropicalization of the space of genus 0 logarithmic stable maps coincides with the space of tropical stable maps, giving a large new collection of examples of faithful tropicalizations for moduli. Moreover, we identify the optimal settings in which the tropicalization of the moduli space of maps is faithful. The Nishinou-Siebert correspondence theorem is shown to be a consequence of this geometric connection between the algebraic and tropical moduli.

“…We obtain a logarithmic map to the original stratum

V (σ_{v})

by composing the underlying map with

\tilde{V} \to V false(σ_{v} false)

, and pushing forward the logarithmic structure using the results of [, Appendix B]. We have constructed the desired map

f_{0}^{v} : C_{v} \to \bar{V} false(σ_{v} false) .

Ranging over all vertices

v

, it is now straightforward to check that these maps

f_{0}^{v}

glue to form a logarithmic map

f_{0} : {scriptC}_{0} \to X .

As argued in [, Section 3.3], by the existence and logarithmic smoothness of

{scriptLSM}_{normalΓ} false(X false)

, we obtain a family

The point

[f_{trop}]

is equivalent to a monoid homomorphism

Q \to {double-struckR}_{⩾ 0}

, and thus a homomorphism

φ : C 〚 Q 〛 \to C 〚 {double-struckR}_{⩾ 0} 〛

. Pulling back along

φ

, we obtain a family

One can check as in [, Section 3.3] that the tropicalization of the family of maps

f

, in the sense of Section 3.4, is precisely

f_{trop}

.…”

Section: The Skeleton Of Scriptlsmmentioning

confidence: 99%

“…is played by the stack of prestable maps to the Artin fan of a logarithmic variety [, Proposition 1.5.1]. The connection between such maps and tropical and nonarchimedean geometry is initiated studied in and will be further studied in .…”

Section: Introductionmentioning

confidence: 99%

Skeletons of stable maps I: rational curves in toric varieties

2017

J. London Math. Soc.

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“…Similarly, each infinite edge is also given a well-defined slope. We will have no reason to describe this type explicitly, so we refrain from describing this association in detail; see [49,48].…”

Section: Replacing Bymentioning

confidence: 99%

Brill-Noether theory for curves of a fixed gonality

Jensen

Forum of Mathematics, Pi

2021

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We prove a generalisation of the Brill-Noether theorem for the variety of special divisors $W^r_d(C)$ on a general curve C of prescribed gonality. Our main theorem gives a closed formula for the dimension of $W^r_d(C)$ . We build on previous work of Pflueger, who used an analysis of the tropical divisor theory of special chains of cycles to give upper bounds on the dimensions of Brill-Noether varieties on such curves. We prove his conjecture, that this upper bound is achieved for a general curve. Our methods introduce logarithmic stable maps as a systematic tool in Brill-Noether theory. A precise relation between the divisor theory on chains of cycles and the corresponding tropical maps theory is exploited to prove new regeneration theorems for linear series with negative Brill-Noether number. The strategy involves blending an analysis of obstruction theories for logarithmic stable maps with the geometry of Berkovich curves. To show the utility of these methods, we provide a short new derivation of lifting for special divisors on a chain of cycles with generic edge lengths, proved using different techniques by Cartwright, Jensen, and Payne. A crucial technical result is a new realisability theorem for tropical stable maps in obstructed geometries, generalising a well-known theorem of Speyer on genus $1$ curves to arbitrary genus.

“…Indeed, this remarkable smoothness property was applied in [29] to show that every tropical stable map arose as the tropicalization of a family of stable maps to the Artin fan.…”

Section: Maps To a Xmentioning

confidence: 99%

“…The earliest realization theorems for superabundant curves are due to Speyer, who observed a subtle combinatorial condition guaranteeing the realizability of superabundant genus 1 tropical curves [31]. While there has been substantial additional work in the intervening years, the general question remains mysterious [7,15,19,[23][24][25]29,34].…”

Section: Introductionmentioning

confidence: 99%

Skeletons of stable maps II: superabundant geometries

2017

Res Math Sci

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We implement new techniques involving Artin fans to study the realizability of tropical stable maps in superabundant combinatorial types. Our approach is to understand the skeleton of a fundamental object in logarithmic Gromov-Witten theory-the stack of prestable maps to the Artin fan. This is used to examine the structure of the locus of realizable tropical curves and derive three principal consequences. First, we prove a realizability theorem for limits of families of tropical stable maps. Second, we extend the sufficiency of Speyer's well-spacedness condition to the case of curves with good reduction. Finally, we demonstrate the existence of liftable genus 1 superabundant tropical curves that violate the well-spacedness condition. BackgroundCentral to the application of tropical techniques to questions in algebraic geometry are so-called lifting theorems. Given a "synthetic" tropical object, such as a weighted balanced polyhedral complex, one must understand whether this object is the tropicalization of an algebraic variety. We deal in this paper with the case of curves. The tropical lifting question in this setting asks, when does an embedded tropical curve in R n arise as the tropicalization of an algebraic curve in a torus over a nonarchimedean field? This question becomes highly nontrivial in the so-called superabundant case and has been the primary obstacle to the application of tropical curve counting techniques in high genus settings. A tropical curve in R n encodes the combinatorial data in a degenerate logarithmic stable map to a toric variety. If the tropical curve is superabundant, i.e., if the tropical deformation space is larger than expected, the obstruction group of this degenerate logarithmic map is nonzero. As a result, such a map may not deform and the tropical curve may fail to be realizable. See Sect. 2.2 for a precise definition of superabundance. The earliest realization theorems for superabundant curves are due to Speyer, who observed a subtle combinatorial condition guaranteeing the realizability of superabundant genus 1 tropical curves [31]. While there has been substantial additional work in the intervening years, the general question remains mysterious [7,15,19,[23][24][25]29,34].In this note, we use recent technical breakthroughs in nonarchimedean geometry and the theory of logarithmic maps to provide a conceptual framework in which the realizability question may be approached. To demonstrate the efficacy of this framework, we use it to give simple proofs of three new results for lifting tropical curves. The same frame- 0123456789().,-: vol