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

Abstract

We bound from below the complexity of the top Chern class λg of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for λg 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 can not 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 λg 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 λg on the moduli of curves after log blow-ups. Contents 1 Introduction 2 λ g in the Chow ring 3 The log Chow ring 4 Relationship with logarithmic geometry 5 The divisor subalgebra of log Chow 6 Pixton's formula for λ g ∈ CH ⋆ (M g ) 7 The bChow ring A The fourth cohomology group of M g B Computations in admcycles

Logarithmic intersections of double ramification cycles

“…The second map fails to be a ring morphism in general. For more details on logarithmic Chow rings and intersection theory see [7,34,47,48].…”

“…We refer the reader to [5, Section 0], [35, Section 0], and [55, Section 5] for more leisurely introductions to the subject of double ramification cycles. For a sampling of the development and application of the theory in a variety of directions, see [4,5,9,10,11,15,18,19,20,25,30,32,33,35,36,47,48,51,57,58,60].…”

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

“…The ideas here are closely related to and informed by the strategy followed in [45]. A related direction is work by Pandharipande, Schmitt, and the first author on the top Chern class of the Hodge bundle on blowups of M g,n [40]. Piecewise polynomials appear here, but the strict transform plays no role in this work, and the formula for the top Chern class of the Hodge bundle becomes the object of study.…”

“…at the ETH Algebraic Geometry and Moduli seminar in 2020 [51]. In the intervening time, piecewise polynomials have appeared in nearby contexts [30,40]. The ring appears to be a natural and interesting invariant of a logarithmic scheme.…”

“…The expression above is "universal" in a similar sense of Aluffi's formula itself. In this toric setting, the content is that the difference between strict and total transforms can be understood completely in terms of a universal formula, whose input is the pushforward to X of classes of strata of V decorated by Chern classes, also known as normally decorated strata classes [40]. 2.7.…”