Deligne\u2019s representation theory in complex rank and objects of integral type

Using Deligne's category Rep(St) we answer questions by B. Kahn and C. Weibel on objects of integral typ

Schur-finite motives and trace identities

We provide a sfficient condition that ensures the nilpotency of endomorphisms universally of trace zero of Schur-finite objects in a category of homological type, i.e., a Q-linear-category with a tensor functor to super vector spaces. We present some applications in the category of motives, where out result generalizes previous results about finite-dimensional objects, in particular by Kimura. We also present some facts which suggest that this mightbe the best generalization possible of this line of proof

Families of curves and variation in moduli

In this paper we study the class of smooth complex projective varieties B such that any modular morphism B -> Mg is constant for any g>1, giving structural properties and examples. Then we investigate the concept of the moduli dimension of a variety B; we bound it by the dimension of the maximal rationally connected quotient of B. In the end, we consider also (generically smooth) families of curves of compact type over rational and elliptic curves