2 results
The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves
- Part of
-
- Journal:
- Compositio Mathematica / Volume 159 / Issue 2 / February 2023
- Published online by Cambridge University Press:
- 14 February 2023, pp. 306-354
- Print publication:
- February 2023
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
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.
On the independence of the generators of tautological rings
- Part of
-
- Journal:
- Compositio Mathematica / Volume 144 / Issue 6 / November 2008
- Published online by Cambridge University Press:
- 01 November 2008, pp. 1497-1503
- Print publication:
- November 2008
-
- Article
-
- You have access
- Export citation
-
We prove that all monomials of κ-classes and ψ-classes are independent in
for all 
. We also give a simple argument for 
in Rl(ℳg) for 
.
for all 
. We also give a simple argument for 
in 
.