Monodromy of Codimension-One Sub-Families of Universal Curves
Suppose that , that and that . The main result is that if is a smooth variety that dominates a codimension subvariety of , the moduli space of -pointed, genus , smooth, projective curves with a level structure, then the closure of the image of the monodromy representation has finite index in . A similar result is proved for codimension families of principally polarized abelian varieties.
1991 Mathematics Subject Classification:Primary 14D05, 14H15; Secondary 14F35, 14F45
Suppose that are integers satisfying , and . Denote the moduli space of smooth complex projective curves of genus with a level structure by and the th power of the universal curve over it by . The moduli space of smooth -pointed smooth projective curves of genus with a level structure is a Zariski open subset of . These will all be regarded as orbifolds. There is a natural monodromy representation
whose image is the level -congruence subgroup of .
The profinite completion of a discrete group will be denoted by . Denote the profinite completion of the integers by . A homomorphism induces a homomorphism .
Suppose that and . If is a dominant morphism from a smooth variety to an irreducible divisor in , then the image of the monodromy representation
has finite index in .
The statement is false when as will be explained in Section LABEL:sec:density. We will prove a stronger version of this theorem (Theorem LABEL:thm:density_curves), in which is replaced by a “suitably generic linear section” of dimension of it. We also prove similar result for abelian varieties (Theorem LABEL:thm:density_ppavs).
Each rational representation of determines a local system over . The theorem implies that when does not contain the trivial representation, the low dimensional cohomology of with coefficients in changes very little when is replaced by a Zariski open subset of or by its generic point.
Suppose that , and . If is a Zariski open subset of , then for all non-trivial, irreducible representations of , the map
induced by the inclusion is an isomorphism when and an injection when .
The groups are known for all when and ; a canonical subspace of it is known [hain:torelli] when when . The resulting computation of the Galois cohomology groups of the absolute Galois group of the function field , when is a number field, is an essential ingredient of the author’s investigation [hain:sec_conj] of rational points of the universal curve over the function field of .
The proof of Theorem 1 easily reduces to the case . Putman [putman:pic] has proved that the Picard group of has rank when . A standard argument using the that fact that is quasi-projective then implies that every irreducible divisor in is ample. A Lefschetz type theorem due to Goresky and MacPherson [smt] implies that when is generic, is an isomorphism. In this case, the result is immediate. The principal difficulty occurs when has self-intersections.
In this case the image of the homomorphism induced by its normalization may have infinite index in , as it does in Figure 1. This issue is addressed by a technical result, Theorem LABEL:thm:strictness, from which Theorem 1 follows directly. It is a hybrid of a “non-abelian strictness theorem” and a Lefschetz-type theorem.
The term non-abelian strictness theorem needs further explanation. Input for one such type of theorem is a diagram