Neutral representations and fields of moduli — Angelo Vistoli (SNS Pisa)

I will present joint work with Giulio Bresciani. Given a perfect field k with algebraic closure k' and a variety X over k', the field of moduli of X is the subfield of k' of elements fixed by elements s of the Galois group of k' over k such that the twist X_s is isomorphic to X. Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and proved that a smooth curve with a marked point is always defined over its field of moduli.

I will review our previous work on a generalization of this result that applies to higher dimensional varieties, and to varieties with additional structures, which is based on the concept of R-singularity; this applies in particular to varieties with a smooth marked point.

Then I will discuss a different technique, which applies much more generally, that of neutral representations. This gives criteria to show that X is defined on its field of moduli, by studying the action of the automorphism group of X on the intrinsically defined cohomology groups of X (for example, the cohomology of the structure sheaf, or the cotangent sheaf). As an application we can show, for example, that if X is a smooth projective curve, whose automorphism group G is abelian of square-free order, and the difference between the genus of X and the genus of X/G is prime with the order of G, then X is defined over its field of moduli.