On the complexity group of stable curves

Busonero, Simone; Melo, Margarida; Stoppino, Lidia

doi:10.1515/advgeom.2011.004

Cited by 5 publications

(4 citation statements)

References 34 publications

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“…where, for some orientation on the graph G X (the choice of which is irrelevant) "δ" and "∂" denote the coboundary and boundary maps defined in subsection 3.2; see [OS79]. See [Ray70], [BLR90], [Lor89], [BMS06] and [BMS11] for more details on the group Φ X .…”

Section: 3mentioning

confidence: 99%

Tropical methods in the moduli theory of algebraic curves

Caporaso¹

2018

Algebraic Geometry: Salt Lake City 2015

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ContentsWe denote by K a field containing k and assumed to be complete with respect to a non-Archimedean valuation, writtenSuch a K is also called, as in [Ber90], a non-Archimedean field. We assume v K induces on k the trivial valuation, k * → 0. We denote by R the valuation ring of K.In this paper any field extension K ′ |K is an extension of non-Archimedean fields so that K ′ will be endowed with a valuation inducing v K .Curves are always assumed to be reduced, projective and having at most nodes as singularities.g ≥ 2 is an integer. G = (V, E) a finite graph (loops and multiple edges allowed), V and E the sets of vertices and edges. We also write V = V (G) and E = E(G). Our graphs will be connected. G = (G, w) denotes a weighted graph. Γ = (G, w, ℓ) denotes a tropical curve, possibly extended. We use the "Kronecker" symbol: for two objects x and y κ x,y := 1 if x = y, 0 otherwise.In drawing graphs we shall denote by a "•" the vertices of weight 0 and by a "•" the vertices of positive weight.Acknowledgements. The paper benefitted from comments and suggestions from Sam Payne, Filippo Viviani, and the referees, to whom I am grateful. Tropical curves2.1. Abstract tropical curves. We begin by defining abstract tropical curves, also known as "metric graphs", following, with a slightly different terminology, [Mik07b], [MZ08], and [BMV11].Definition 2.1.1. A pure tropical curve is a pair Γ = (G, ℓ) such that G = (V, E) is a graph and ℓ : E → R >0 is a length function on the edges.More generally, a (weighted) tropical curve is a triple Γ = (G, ℓ, w) such that G and ℓ are as above, and w : V → Z ≥0 is a weight function on the vertices.

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Section: 3mentioning

confidence: 99%

Tropical methods in the moduli theory of algebraic curves

Caporaso¹

2018

Algebraic Geometry: Salt Lake City 2015

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“…The equivalence classes of multidegrees that sum up to d are denoted by Note that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Delta _X:=\Delta _X^0$\end{document} is a finite group and that each \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Delta _X^d$\end{document} is a torsor under Δ X . The group Δ X is known in the literature under many different names (see 8 and the references therein); we will follow the terminology introduced in 9 and call it the degree class group of X .…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

Fine compactified Jacobians

Melo

Viviani

2012

Mathematische Nachrichten

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Abstract. To every singular reduced projective curve X one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of a finite number of copies of the generalized Jacobian of X. We investigate the geometric properties of fine compactified Jacobians focusing on curves having locally planar singularities. We give examples of nodal curves admitting non isomorphic (and even non homeomorphic over the field of complex numbers) fine compactified Jacobians. We study universal fine compactified Jacobians, which are relative fine compactified Jacobians over the semiuniversal deformation space of the curve X. Finally, we investigate the existence of twisted Abel maps with values in suitable fine compactified Jacobians.

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“…cit., the authors also describe a stratification of the whole compactification J E,σ f in terms of Néron models of partial normalizations of the special fiber of f . In this case, the fiber of N (J(X U )) over the special fiber of B is known to be isomorphic to a disjoint sum of copies of the generalized Jacobian of the special fiber of f , so the result acquires a combinatorial flavor (indeed Néron models are in this case determined by the so-called degree class group, which is defined using the dual graph of the special fiber of f and that has many interesting incarnations in the literature: see for instance the introduction of [BMS11] for an account of these different appearances).…”

Section: Line Bundles Loci In Compactifiedmentioning

confidence: 99%

Universal Néron models for Jacobians of curves with marked points

Melo

2016

Boll Unione Mat Ital

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We consider the problem of constructing universal Néron models for families of curves with sections. By applying a construction of the author of universal compactified Jacobians over the moduli stack of reduced curves with markings and a result by J. Kass, we get a positive answer for smooth families of curves with planar singularities over Dedekind schemes.Question 1.1. Let M g,n be the moduli stack of reduced curves of given genus g and with n distinct markings. Is there a modular algebraic stack N g,n endowed with a map onto M g,n such that N g,n is a universal Néron model for the Jacobians of families of curves in M g,n ? Moreover, can we describe a modular compactification of N g,n over M g,n ?By a universal Néron model we mean an algebraic stack N g,n endowed with a map π onto M g,n such that given a family of curves (f : X → B; σ 1 , . . . , σ n ) ∈ M g,n (B) over a Dedekind scheme B, which is smooth over a dense open subset U ⊆ B, the pullback of N g,n → M g,n via the moduli map of the family µ f : B → M g,n is isomorphic to the Néron model of J(X U ) over B:A different possibility for extending J(X U ) over B is to take a compactified Jacobian of the family f . This is a proper model of J(X U ) → U and in the case when f has non-integral 1991 Mathematics Subject Classification. Primary 14H10; Secondary 14H40.

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On the complexity group of stable curves

Cited by 5 publications

References 34 publications

Tropical methods in the moduli theory of algebraic curves

Tropical methods in the moduli theory of algebraic curves

Fine compactified Jacobians

Universal Néron models for Jacobians of curves with marked points

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