Pixton’s formula and Abel–Jacobi theory on the Picard stack

Bae, Younghan; Holmes, David; Pandharipande, Rahul; Schmitt, Johannes; Schwarz, Rosa

doi:10.4310/acta.2023.v230.n2.a1

Acta Mathematica

2023

DOI: 10.4310/acta.2023.v230.n2.a1

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Pixton’s formula and Abel–Jacobi theory on the Picard stack

Younghan Bae

David Holmes

Rahul Pandharipande

et al.

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2024

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Cited by 4 publications

References 60 publications

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Stability conditions for line bundles on nodal curves

Pagani,

Tommasi

2024

Forum of Mathematics, Sigma

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We introduce the abstract notion of a smoothable fine compactified Jacobian of a nodal curve, and of a family of nodal curves whose general element is smooth. Then we introduce the combinatorial notion of a stability assignment for line bundles and their degenerations. We prove that smoothable fine compactified Jacobians are in bijection with these stability assignments. We then turn our attention to fine compactified universal Jacobians – that is, fine compactified Jacobians for the moduli space $\overline {\mathcal {M}}_g$ of stable curves (without marked points). We prove that every fine compactified universal Jacobian is isomorphic to the one first constructed by Caporaso, Pandharipande and Simpson in the nineties. In particular, without marked points, there exists no fine compactified universal Jacobian unless $\gcd (d+1-g, 2g-2)=1$ .

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Stability conditions for line bundles on nodal curves

Pagani,

Tommasi

2024

Forum of Mathematics, Sigma

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Pluricanonical cycles and tropical covers

Cavalieri,

Markwig,

Ranganathan

2024

Trans. Amer. Math. Soc.

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We extract a system of numerical invariants from logarithmic intersection theory on pluricanonical double ramification cycles, and show that these invariants exhibit a number of properties that are enjoyed by double Hurwitz numbers. Among their properties are (i) the numbers can be efficiently calculated by counts of tropical curves with a modified balancing condition, (ii) they are piecewise polynomial in the entries of the ramification vector, and (iii) they are matrix elements of operators on the Fock space. The numbers are extracted from the logarithmic double ramification cycle, which is a lift of the standard double ramification cycle to a blowup of the moduli space of curves. The blowup is determined by tropical geometry. We show that the traditional double Hurwitz numbers are intersections of the refined cycle with the cohomology class of a piecewise polynomial function on the tropical moduli space of curves. This perspective then admits a natural, combinatorially motivated, generalization to the pluricanonical setting. Tropical correspondence results for the new invariants lead immediately to the structural results for these numbers.

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A case study of intersections on blowups of the moduli of curves

Molcho,

Ranganathan

2024

Alg. Number Th.

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We explain how logarithmic structures select principal components in an intersection of schemes. These manifest in Chow homology and can be understood using strict transforms under logarithmic blowups. Our motivation comes from Gromov-Witten theory. The toric contact cycles in the moduli space of curves parameterize curves that admit a map to a fixed toric variety with prescribed contact orders. We show that they are intersections of virtual strict transforms of double ramification cycles in blowups of the moduli space of curves. We supply a calculation scheme for the virtual strict transforms, and deduce that toric contact cycles lie in the tautological ring of the moduli space of curves. This is a higher-dimensional analogue of a result of Faber and Pandharipande. The operational Chow rings of Artin fans play a basic role, and are shown to be isomorphic to rings of piecewise polynomials on associated cone complexes. The ingredients in our analysis are Fulton's blowup formula, Aluffi's formulas for Segre classes of monomial schemes, piecewise polynomials, and degeneration methods. A model calculation in toric intersection theory is treated without logarithmic methods and may be read independently. MSC2020: 14A21, 14H10.

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Pixton’s formula and Abel–Jacobi theory on the Picard stack

Cited by 4 publications

References 60 publications

Stability conditions for line bundles on nodal curves

Stability conditions for line bundles on nodal curves

Pluricanonical cycles and tropical covers

A case study of intersections on blowups of the moduli of curves

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