Ample Pairs

We show that the ample degree of a stable theory with trivial forking is preserved when we consider the corresponding theory of belles paires, if it exists. This result also applies to the theory of $H$-structures of a trivial theory of rank $1$.Comment: Research partially supported by the program MTM2014-59178-P. The second author conducted research with support of the programme ANR-13-BS01-0006 Valcomo. The third author would like to thank the European Research Council grant 33882

Ample Pairs

We show that the ample degree of a stable theory with trivial forking is preserved when we consider the corresponding theory of belles paires, if it exists. This result also applies to the theory of $H$-structures of a trivial theory of rank $1$.Comment: Research partially supported by the program MTM2014-59178-P. The second author conducted research with support of the programme ANR-13-BS01-0006 Valcomo. The third author would like to thank the European Research Council grant 33882

Fields and Fusions: Hrushovski constructions and their definable groups

An overview is given of the various expansions of fields and fusions of strongly minimal sets obtained by means of Hrushovski's amalgamation method, as well as a characterization of the groups definable in these structures

Weak one-basedness

We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterises linearity in the setting of geometric þ-rank 1 structures and that lovely pairs of weakly one-based geometric þ-rank 1 struc- tures are weakly one-based with respect to þ-independence. We also study geometries arising from infinite dimensional vector spaces over division rings

Some Definability Results in Abstract Kummer Theory

Let $S$ be a semiabelian variety over an algebraically closed field, and let $X$ be an irreducible subvariety not contained in a coset of a proper algebraic subgroup of $S$. We show that the number of irreducible components of $[n]^{-1}(X)$ is bounded uniformly in $n$, and moreover that the bound is uniform in families $X_t$. We prove this by purely Galois-theoretic methods. This proof applies in the more general context of divisible abelian groups of finite Morley rank. In this latter context, we deduce a definability result under the assumption of the Definable Multiplicity Property (DMP). We give sufficient conditions for finite Morley rank groups to have the DMP, and hence give examples where our definability result holds.Comment: 21 pages; minor notational fixe