The Automorphism Group of the Combinatorial Geometry of an Algebraically Closed Field

Throughout this paper L will be an algebraically closed field and K an algebraically closed subfield such that the transcendence rank of L over K is at least 3. In [4] we considered the pregeometry G(L/K) = (L,clK), where, for Y a subset of L, the closure clK( Y) is the algebraic closure in L of the subfield generated by K and Y. In this paper we shall be concerned mainly with the geometry G(L/K) associated to this: so that the point set P ofG(L/K) is the set of algebraically closed subfields of L which contain K and are of transcendence rank 1 over K, and the closure clK(Z) of Z £ P is the set of points which are contained in clK(JJ Z) (see [4, 1.1] for more details). One of the aims of [4] was to compare these geometries with the more familiar projective geometries (where the pregeometry is given by linear closure in a vector space and the corresponding geometry is the projective geometry). In this paper we pursue this theme further and obtain results on the automorphism group of G(L/K), and the recovery of L from G(L/K) (so these are analogues of two classical results from projective geometry: the Fundamental Theorem and the Coordinatisation Procedure). Let Aut(L){K} be the group of automorphisms of L which fix AT setwise. It is clear that any ge Aut(£){K) is an automorphism of the pregeometry G(L/K), and therefore induces an automorphism %(g) of the geometry G(L/K) (given by x(g)(c\K(x)) = c\K(gx) for all xeL\K). The map xAut(L){K}-^ Aut(G(L/K)) is a homomorphism. Daniel Lascar asked whether % is surjective. Our main result is a partial answer to this.