인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

We describe the closure of the strata of abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne-Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.

Compactification of strata of abelian differentials

Gromov–Witten invariants have been constructed to be deformation invariant, but their behavior under other transformations is subtle. We show that logarithmic Gromov–Witten invariants are also invariant under appropriately defined logarithmic modifications.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/B203EFA803C7B10BF56F386A1C65CC7B/S0010437X17007667a.pdf/div-class-title-birational-invariance-in-logarithmic-gromov-witten-theory-div.pdf

Birational invariance in logarithmic Gromov–Witten theory

We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of $\mathbb{T}$-points this reduces to Kajiwara-Payne's extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of $\mathbb{T}$-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.

/pdf/equations-of-tropical-varieties-2ah40xv7op.pdf

Equations of tropical varieties

We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves.Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.

/pdf/a-moduli-stack-of-tropical-curves-4zo0foo4kc.pdf

A moduli stack of tropical curves

We survey a collection of closely related methods for generalizing fans of toric varieties, include skeletons, Kato fans, Artin fans, and polyhedral cone complexes, all of which apply in the wider context of logarithmic geometry. Under appropriate assumptions these structures are equivalent, but their different realizations have provided for surprisingly disparate uses. We highlight several current applications and suggest some future possibilities.

Skeletons and Fans of Logarithmic Structures

Hassett's moduli spaces of weighted stable curves form an important class of alternate modular compactifications of the moduli space of smooth curves with marked points. In this article we define a tropical analogue of these moduli spaces and show that the naive set-theoretic tropicalization map can be identified with a natural deformation retraction onto the non-Archimedean skeleton. This result generalizes work of Abramovich, Caporaso, and Payne treating the Deligne-Knudsen-Mumford compactification of the moduli space of smooth curves with marked points. We also study tropical analogues of the tautological maps, investigate the dependence of the tropical moduli spaces on the weight data, and consider the example of Losev-Manin spaces.

Tropical geometry of moduli spaces of weighted stable curves

For a complex toric variety $X$ the logarithmic absolute value induces a natural retraction of $X$ onto the set of its non-negative points and this retraction can be identified with a quotient of $X(\mathbb{C})$ by its big real torus. We prove an analogous result in the non-Archimedean world: The Kajiwara-Payne tropicalization map is a non-Archimedean analytic stack quotient of $X^{an}$ by its big affinoid torus. Along the way, we provide foundations for a geometric theory of non-Archimedean analytic stacks, particularly focussing on analytic groupoids and their quotients, the process of analytification, and the underlying topological spaces of analytic stacks.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2017/0024/0004/MRL-2017-0024-0004-a012.pdf

Tropicalization is a non-Archimedean analytic stack quotient

We contribute to the foundations of tropical geometry with a view towards formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves. Our approach to tropical geometry permits tropical moduli problems---moduli of curves or otherwise---to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.

/pdf/a-moduli-stack-of-tropical-curves-1rsx25xgt6.pdf

Martin Ulirsch

Papers

A moduli stack of tropical curves

Skeletons and Fans of Logarithmic Structures

Tropical geometry of moduli spaces of weighted stable curves

Tropicalization is a non-Archimedean analytic stack quotient

A moduli stack of tropical curves.