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

Molcho, Sam; Ranganathan, Dhruv

doi:10.2140/ant.2024.18.1767

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

Sam Molcho and Dhruv Ranganathan

Vol. 18 (2024), No. 10, 1767–1816

DOI: 10.2140/ant.2024.18.1767

Abstract

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.

Keywords

logarithmic geometry, double ramification cycle, Gromov–Witten theory

Mathematical Subject Classification

Primary: 14A21, 14H10

Milestones

Received: 14 July 2021

Revised: 2 June 2023

Accepted: 31 October 2023

Published: 7 October 2024

Authors

Sam Molcho
Department of Mathematics ETH Zürich Zürich Switzerland

Dhruv Ranganathan
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge United Kingdom

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Vol. 18, No. 10, 2024