The singularities and birational geometry of the compactified universal Jacobian

Casalaina-Martin, Sebastian; Kass, Jesse Leo; Viviani, Filippo

doi:10.14231/ag-2017-020

Cited by 8 publications

(20 citation statements)

References 18 publications

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“…Let

[(X, I)]

be a point in

{\bar{scriptJ}}_{μ, g}

and

(Γ, E, D)

be the dual graph of

(X, I)

. It follows from Lemma and [, Equation 8.4] that, up to choosing an orientation of

normalΓ

{\overset{O}{true}}_{{\bar{J}}_{μ, g} false[(C, I) false]} = A \otimes k false[[W_{1}, \dots, W_{3 g - 2 - | E (Γ) |}, Z_{1}, \dots, Z_{g - | E |}] false],

where

A : = ⨂_{e \in E} \frac{k [false[X_{e}, Y_{e}, T_{e} false]]}{⟨ X_{e} Y_{e} - T_{e} ⟩} \otimes (⨂_{e \in E false(normalΓ false) ∖ scriptE} k [false[T_{e} false]]) .

…”

Section: The Skeleton Of the Esteves' Universal Jacobianmentioning

confidence: 99%

“…The local ring at

[X]

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

{\overset{O}{true}}_{{\bar{M}}_{g, 1}, false[X false]} = ⨂_{e \in E (Γ)} k false[[T_{e}] false] \otimes k false[[W_{1}, \dots, W_{3 g - 2 - | E (Γ) |}] false],

and the forgetful map

π : {\bar{J}}_{μ, g} \to {\bar{M}}_{g, 1}

is induced locally at

[(X, I)]

by the ring homomorphism

π^{#} : {\overset{O}{true}}_{{\bar{M}}_{g, 1}, false[X false]} \to {\overset{O}{true}}_{{\bar{J}}_{μ, g}, false[(X, I) false]}

given by

T_{e} \mapsto T_{e}

and

W_{i} \mapsto W_{i}

(see [, equation 7.17]).…”

Section: The Skeleton Of the Esteves' Universal Jacobianmentioning

confidence: 99%

“…We note that, although the results in [15] hold for J d,g , their arguments are based on deformation theory and can be naturally extended to the pointed case needed to analyze J d,g,1 .…”

Section: The Stratification Of J μGmentioning

confidence: 99%

“…Since J μ,g and M g,n is proper, then J Let (Γ, E, D) be the dual graph of [(X, I)]. Recall the notation in equation (15). We define a length function : E(Γ E ) → R >0 given by (e) = ν R (ψ # (T e )) for e ∈ E(Γ) \ E, while (e ) = ν R (ψ # (X e )) and (e ) = ν R (ψ # (Y e )) for e ∈ E, where e and e are the edges of Γ E over e ∈ E(Γ) corresponding to the source and target of e, respectively.…”

Section: The Tropicalization Of J μGmentioning

confidence: 99%

“…Finally, thanks to equations (15) and (16), we have that trop Mg,1 • π an (ψ) is the tropical curve (Γ, ) where (e) = ν R (ψ # T e ). On the other hand, π trop • trop J μ,g (ψ) is the tropical curve (Γ E , ) where is as in Definition 6.7.…”

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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“…Let

[(X, I)]

be a point in

{\bar{scriptJ}}_{μ, g}

and

(Γ, E, D)

be the dual graph of

(X, I)

. It follows from Lemma and [, Equation 8.4] that, up to choosing an orientation of

normalΓ

{\overset{O}{true}}_{{\bar{J}}_{μ, g} false[(C, I) false]} = A \otimes k false[[W_{1}, \dots, W_{3 g - 2 - | E (Γ) |}, Z_{1}, \dots, Z_{g - | E |}] false],

where

A : = ⨂_{e \in E} \frac{k [false[X_{e}, Y_{e}, T_{e} false]]}{⟨ X_{e} Y_{e} - T_{e} ⟩} \otimes (⨂_{e \in E false(normalΓ false) ∖ scriptE} k [false[T_{e} false]]) .

…”

Section: The Skeleton Of the Esteves' Universal Jacobianmentioning

confidence: 99%

“…The local ring at

[X]

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

{\overset{O}{true}}_{{\bar{M}}_{g, 1}, false[X false]} = ⨂_{e \in E (Γ)} k false[[T_{e}] false] \otimes k false[[W_{1}, \dots, W_{3 g - 2 - | E (Γ) |}] false],

and the forgetful map

π : {\bar{J}}_{μ, g} \to {\bar{M}}_{g, 1}

is induced locally at

[(X, I)]

by the ring homomorphism

π^{#} : {\overset{O}{true}}_{{\bar{M}}_{g, 1}, false[X false]} \to {\overset{O}{true}}_{{\bar{J}}_{μ, g}, false[(X, I) false]}

given by

T_{e} \mapsto T_{e}

and

W_{i} \mapsto W_{i}

(see [, equation 7.17]).…”

Section: The Skeleton Of the Esteves' Universal Jacobianmentioning

confidence: 99%

“…We note that, although the results in [15] hold for J d,g , their arguments are based on deformation theory and can be naturally extended to the pointed case needed to analyze J d,g,1 .…”

Section: The Stratification Of J μGmentioning

confidence: 99%

Section: The Tropicalization Of J μGmentioning

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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On the unirationality of moduli spaces of pointed curves

Keneshlou

Tanturri

2021

Math. Z.

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We show that $$\mathcal {M}_{g,n}$$ M g , n , the moduli space of smooth curves of genus g together with n marked points, is unirational for $$g=12$$ g = 12 and $$2 \le n\le 4$$ 2 ≤ n ≤ 4 and for $$g=13$$ g = 13 and $$1 \le n \le 3$$ 1 ≤ n ≤ 3 , by constructing suitable dominant families of projective curves in $$\mathbb {P}^1 \times \mathbb {P}^2$$ P 1 × P 2 and $$\mathbb {P}^3$$ P 3 respectively. We also exhibit several new unirationality results for moduli spaces of smooth curves of genus g together with n unordered points, establishing their unirationality for $$g=11, n=7$$ g = 11 , n = 7 and $$g=12, n =5,6$$ g = 12 , n = 5 , 6 .

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The Picard group of the universal moduli space of vector bundles on stable curves

Fringuelli

2018

Advances in Mathematics

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The singularities and birational geometry of the compactified universal Jacobian

Cited by 8 publications

References 18 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

On the unirationality of moduli spaces of pointed curves

The Picard group of the universal moduli space of vector bundles on stable curves

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