The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves. Issue 2 (14th February 2023)

• Record Type: Journal Article
• Authors: Molcho, S.; Pandharipande, R.; Schmitt, J.
• Is Part Of: Compositio mathematica. Volume 159:Issue 2(2023)
• Page Start: 306
• Page End: 354
• Publication Date: 2023-02-14
• Subjects: moduli space of curves -- Hodge bundle -- tautological rings -- logarithmic intersection theory -- computer algebra; 14H10 -- 14C17 -- 14A21 -- 14K10 -- 14-04; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X22007874
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : 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.