The logarithmic Picard group and its tropicalization

Molcho, Samouil; Wise, Jonathan

doi:10.1112/s0010437x22007527

Cited by 16 publications

(14 citation statements)

References 46 publications

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“…Each approach comes with its own advantages and limitations, and in each case there is a large body of literature. See, for example, [4,7,17,[23][24][25] and [2,22].…”

Section: Previous Resultsmentioning

confidence: 99%

A Clifford inequality for semistable curves

Christ

2022

Math. Z.

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Let X be a semistable curve and L a line bundle whose multidegree is uniform, i.e., in the range between those of the structure sheaf and the dualizing sheaf of X. We establish an upper bound for $$h^0(X,L)$$ h 0 ( X , L ) , which generalizes the classic Clifford inequality for smooth curves. The bound depends on the total degree of L and connectivity properties of the dual graph of X. It is sharp, in the sense that on any semistable curve there exist line bundles with uniform multidegree that achieve the bound.

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“…Each approach comes with its own advantages and limitations, and in each case there is a large body of literature. See, for example, [4,7,17,[23][24][25] and [2,22].…”

Section: Previous Resultsmentioning

confidence: 99%

A Clifford inequality for semistable curves

Christ

2022

Math. Z.

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“…Holmes, D. Ranganathan, and J. Wise have suggested that the -formula (2) should extend over the moduli of curves in some form in log geometry (based on their understanding of the logarithmic Picard stack [MW22]). Our initial motivation here was to study geometric obstructions to such an extension.…”

Section: Acknowledgementsmentioning

confidence: 99%

The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves

Molcho¹,

Pandharipande²,

Schmitt³

2023

Compositio Math.

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We bound from below the complexity of the top Chern class $\lambda _g$ of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for $\lambda _g$ in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also cannot be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove $\lambda _g$ lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for $\lambda _g$ on the moduli of curves after log blow-ups.

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“…This can only happen if 𝑣 𝑒 𝑝 ,𝑒 ⊂ 𝑍 (recall that 𝑣 𝑒 , 𝑝 ,𝑒 ∈ 𝑍), and in this case, equality holds in (24). □ Proof of item (2) of Theorem 5.9.…”

Section: Polystability On Tropical Curvesmentioning

confidence: 99%

“…We also construct polyhedral decompositions of the Jacobian

J(\operatorname{\Gamma })

of a tropical curve

\operatorname{\Gamma }

. Polyhedral decompositions of the tropical Jacobian are closely related to toroidal compactifications of the Jacobian of a curve (see [15, 24, 25]). An interesting decomposition is studied in [4] in degree

g

through break divisors.…”

Section: Introductionmentioning

confidence: 99%

A universal tropical Jacobian over Mgtrop$M_g^{\rm {trop}}$

Abreu

Andria

Pacini

et al. 2022

Journal of London Math Soc

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We introduce and study polystable divisors on a tropical curve, which are the tropical analogue of polystable torsion‐free rank‐1 sheaves on a nodal curve. We construct a universal tropical Jacobian over the moduli space of tropical curves of genus g$g$. This space parametrizes equivalence classes of tropical curves of genus g$g$ together with a μ$\mu$‐polystable divisor, and can be seen as a tropical counterpart of Caporaso universal Picard scheme. We describe polyhedral decompositions of the Jacobian of a tropical curve via polystable divisors, relating them with other known polyhedral decompositions.

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The logarithmic Picard group and its tropicalization

Cited by 16 publications

References 46 publications

A Clifford inequality for semistable curves

A Clifford inequality for semistable curves

The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves

A universal tropical Jacobian over Mgtrop$M_g^{\rm {trop}}$

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