The stability space of compactified universal Jacobians

Kass, Jesse Leo; Pagani, Nicola

doi:10.1090/tran/7724

Cited by 16 publications

(41 citation statements)

References 35 publications

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“…We have the canonical polarization , given by

μ_{normalΓ} false(v false) = \frac{d (2 w (v) - 2 + val (v))}{2 g - 2} .

We also have a polarization concentrated at the marked vertex

v_{0}

normalΓ

, given by

μ_{normalΓ} false(v false) = \{\begin{matrix} left \begin{matrix} 0, & if v \neq v_{0}; \\ d, & if v = v_{0} . \end{matrix} \end{matrix}

Note that any linear combination of the above universal polarizations gives rise to a universal genus‐

g

polarization of degree

d

. These are the only universal polarizations one can consider on

{Graph}_{g, 1}

: this is proved in [, Section 5].…”

Section: Quasistability On Graphsmentioning

confidence: 99%

“…For every proper subcurve

Y

X

, we set

μ false(Y false) : = \sum_{X_{i} \subset Y} μ_{i} .

The notion of a degree‐

d

polarization on a curve can also be given by a vector bundle

scriptF

X

such that

deg (F) = - d \cdot rank (F)

. In this case,

μ_{i} = - deg false(scriptF |_{X_{i}} false) / rank false(scriptF false)

for every

i = 1, \dots, m

(see [, Section 1.2; , Remark 4.6]).…”

Section: The Skeleton Of the Esteves' Universal Jacobianmentioning

confidence: 99%

“…For every scheme

S

, we have

{\bar{J}}_{μ, g} false(S false) = \frac{({} false(π, σ, scriptI false); \begin{matrix} π : X \to S is a family of stable pointed genus-g curves, \\ I is a false(σ, μ false) -quasistable torsion-free rank-1 sheaf on X \end{matrix})}{\sim},

where

false(π_{1}, σ_{1}, I_{1} false) \sim false(π_{2}, σ_{2}, I_{2} false)

if there exist a

S

‐isomorphism

f : {scriptX}_{1} \to {scriptX}_{2}

and an invertible sheaf

scriptL

S

, such that

σ_{2} = f \circ σ_{1}

and

{scriptI}_{1} ≅ f^{*} {scriptI}_{2} \otimes π_{1}^{*} L

. We refer to [, Corollary 4.4, Remark 4.6; , Theorems A and B] for more details on the stack

{\bar{scriptJ}}_{μ, g}

. In what follows we will consider the universal compactified Jacobian

{\bar{scriptJ}}_{μ, g}

where

μ

is the canonical polarization (recall Example ).…”

Section: The Skeleton Of the Esteves' Universal Jacobianmentioning

confidence: 99%

“…Recently, Caporaso , Melo , and Kass and Pagani carried out a stacky version of theses Jacobians. In particular, Melo constructed a Deligne–Mumford stack

{\bar{scriptJ}}_{μ, g}

over the moduli space

{\bar{scriptM}}_{g, 1}

of stable pointed genus‐

g

curves, following the work of Esteves .…”

Section: Introductionmentioning

confidence: 99%

“…Finally, Altman and Kleiman [3] and Esteves [19] constructed a compactified relative Jacobian for families of reduced curves, without using GIT. This compactification employs the notion of quasistability for torsion-free rank-1 sheaves with respect to a polarization, and depends on the choice of a section of the family of curves.Recently, Caporaso [10], Melo [24,25], and Kass and Pagani [22] carried out a stacky version of theses Jacobians. In particular, Melo [25] constructed a Deligne-Mumford stack J μ,g over the moduli space M g,1 of stable pointed genus-g curves, following the work of Esteves [19].…”

mentioning

confidence: 99%

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The universal tropical Jacobian and the skeleton of the Esteves' universal Jacobian

Abreu

Pacini

2020

Proc. London Math. Soc.

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For each universal genus‐g polarization μ of degree d, we construct a universal tropical Jacobian Jμ,gtrop as a generalized cone complex over the moduli space of stable pointed genus‐g tropical curves. We show several properties of the space Jμ,gtrop. In particular, we prove that the natural compactification of Jμ,gtrop is the tropicalization of the Esteves' compactified universal Jacobian over the moduli space of stable pointed genus‐g curves.

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“…We have the canonical polarization , given by

μ_{normalΓ} false(v false) = \frac{d (2 w (v) - 2 + val (v))}{2 g - 2} .

We also have a polarization concentrated at the marked vertex

v_{0}

normalΓ

, given by

μ_{normalΓ} false(v false) = \{\begin{matrix} left \begin{matrix} 0, & if v \neq v_{0}; \\ d, & if v = v_{0} . \end{matrix} \end{matrix}

Note that any linear combination of the above universal polarizations gives rise to a universal genus‐

g

polarization of degree

d

. These are the only universal polarizations one can consider on

{Graph}_{g, 1}

: this is proved in [, Section 5].…”

Section: Quasistability On Graphsmentioning

confidence: 99%

“…For every proper subcurve

Y

X

, we set

μ false(Y false) : = \sum_{X_{i} \subset Y} μ_{i} .

The notion of a degree‐

d

polarization on a curve can also be given by a vector bundle

scriptF

X

such that

deg (F) = - d \cdot rank (F)

. In this case,

μ_{i} = - deg false(scriptF |_{X_{i}} false) / rank false(scriptF false)

for every

i = 1, \dots, m

(see [, Section 1.2; , Remark 4.6]).…”

Section: The Skeleton Of the Esteves' Universal Jacobianmentioning

confidence: 99%

“…For every scheme

S

, we have

{\bar{J}}_{μ, g} false(S false) = \frac{({} false(π, σ, scriptI false); \begin{matrix} π : X \to S is a family of stable pointed genus-g curves, \\ I is a false(σ, μ false) -quasistable torsion-free rank-1 sheaf on X \end{matrix})}{\sim},

where

false(π_{1}, σ_{1}, I_{1} false) \sim false(π_{2}, σ_{2}, I_{2} false)

if there exist a

S

‐isomorphism

f : {scriptX}_{1} \to {scriptX}_{2}

and an invertible sheaf

scriptL

S

, such that

σ_{2} = f \circ σ_{1}

and

{scriptI}_{1} ≅ f^{*} {scriptI}_{2} \otimes π_{1}^{*} L

. We refer to [, Corollary 4.4, Remark 4.6; , Theorems A and B] for more details on the stack

{\bar{scriptJ}}_{μ, g}

. In what follows we will consider the universal compactified Jacobian

{\bar{scriptJ}}_{μ, g}

where

μ

is the canonical polarization (recall Example ).…”

Section: The Skeleton Of the Esteves' Universal Jacobianmentioning

confidence: 99%

“…Recently, Caporaso , Melo , and Kass and Pagani carried out a stacky version of theses Jacobians. In particular, Melo constructed a Deligne–Mumford stack

{\bar{scriptJ}}_{μ, g}

over the moduli space

{\bar{scriptM}}_{g, 1}

of stable pointed genus‐

g

curves, following the work of Esteves .…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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The universal tropical Jacobian and the skeleton of the Esteves' universal Jacobian

Abreu

Pacini

2020

Proc. London Math. Soc.

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Pullbacks of universal Brill–Noether classes via Abel–Jacobi morphisms

Pagani

Ricolfi

Zelm

2020

Mathematische Nachrichten

Self Cite

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Following Mumford and Chiodo, we compute the Chern character of the derived pushforward ch (• * ()) , for an arbitrary element of the Picard group of the universal curve over the moduli stack of stable marked curves. This allows us to express the pullback of universal Brill-Noether classes via Abel-Jacobi sections to the compactified universal Jacobians, for all compactifications such that the section is a well-defined morphism.

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A non-Archimedean analogue of Teichmüller space and its tropicalization

Ulirsch

2021

Sel. Math. New Ser.

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In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichmüller space$$\overline{{{\mathcal {T}}}}_g$$ T ¯ g whose points are pairs consisting of a stable projective curve over a non-Archimedean field and a Teichmüller marking of the topological fundamental group of its Berkovich analytification. This construction is closely related to and inspired by the classical construction of a non-Archimedean Schottky space for Mumford curves by Gerritzen and Herrlich. We argue that the skeleton of non-Archimedean Teichmüller space is precisely the tropical Teichmüller space introduced by Chan–Melo–Viviani as a simplicial completion of Culler–Vogtmann Outer space. As a consequence, Outer space turns out to be a strong deformation retract of the locus of smooth Mumford curves in $$\overline{{\mathcal {T}}}_g$$ T ¯ g .

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The stability space of compactified universal Jacobians

Cited by 16 publications

References 35 publications

The universal tropical Jacobian and the skeleton of the Esteves' universal Jacobian

The universal tropical Jacobian and the skeleton of the Esteves' universal Jacobian

Pullbacks of universal Brill–Noether classes via Abel–Jacobi morphisms

A non-Archimedean analogue of Teichmüller space and its tropicalization

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