Oda’s problem, which deals with the fixed field of the universal monodromy representation of moduli spaces of curves and its independence with respect to the topological data, is a central question of anabelian arithmetic geometry. This paper emphasizes the stack nature of this problem by establishing the independence of monodromy fields with respect to finer special loci data of curves with symmetries, which we show provides a new proof of Oda’s prediction.

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.

The aim of this paper is to study all the natural first steps of the minimal model program for the moduli space of stable pointed curves. We prove that they admit a modular interpretation, and we study their geometric properties. As a particular case, we recover the first few Hassett–Keel log canonical models. As a by-product, we produce many birational morphisms from the moduli space of stable pointed curves to alternative modular projective compactifications of the moduli space of pointed curves.

Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper, we study affine-related properties of strata of $k$-differentials on smooth curves which parameterize sections of the $k$th power of the canonical line bundle with prescribed orders of zeros and poles. We show that if there is a prescribed pole of order at least $k$, then the corresponding stratum does not contain any complete curve. Moreover, we explore the amusing question whether affine invariant manifolds arising from Teichmüller dynamics are affine varieties, and confirm the answer for Teichmüller curves, Hurwitz spaces of torus coverings, hyperelliptic strata as well as some low genus strata.

Suppose that $K$ is a subfield of $\mathbb{C}$ for which the $\ell$-adic cyclotomic character has infinite image. Suppose that $C$ is a curve of genus $g\geq3$ defined over $K$, and that $\xi$ is a $K$-rational point of $C$. This paper considers the relation between the actions of the mapping class group of the pointed topological curve $(C^{\mathrm{an}},\xi)$ and the absolute Galois group $G_K$ of $K$ on the $\ell$-adic prounipotent fundamental group of $(C^{\mathrm{an}},\xi)$. A close relationship is established between

1. the image of the absolute Galois group of $K$ in the automorphism group of the $\ell$-adic unipotent fundamental group of $C$; and

2. the $\ell$-adic Galois cohomology classes associated to the algebraic $1$-cycle $C- C^{-}$ in the Jacobian of $C$, and to the algebraic $0$-cycle $(2g-2)\xi-K_C$ in $C$.

AMS 2000 Mathematics subject classification: Primary 11G30. Secondary 14H30; 12G05; 14C25; 14G32

Each finite dimensional irreducible rational representation V of the symplectic group Sp$_2g$(Q) determines a generically defined local system V over the moduli space M$_g$ of genus g smooth projective curves. We study H$^2$ (M$_g$; V) and the mixed Hodge structure on it. Specifically, we prove that if g [ges ] 6, then the natural map IH$^2$(M˜$_g$; V) → H$^2$(M$_g$; V) is an isomorphism where M˜$_g$ is the Satake compactification of M$_g$. Using the work of Saito we conclude that the mixed Hodge structure on H$^2$(M$_g$; V) is pure of weight 2+r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H$^2$(M$_g$; V) for 3 [les ] g < 6. Results of this article can be applied in the study of relations in the Torelli group T$_g$.