Projective geometries of algebraically closed fields of characteristic zero (Q1210352)

For \(K\) an algebraically closed field of characteristic 0 and \(k\) an algebraically closed subfield of \(K\), let \(G\) denote the geometry whose points are the algebraically closed subfields of \(K\) of transcendence degree 1 over \(k\), with collinearity defined by algebraic dependence. It is known that any (finite dimensional) projective subgeometry of \(G\) is Desarguesian. In the present paper it is shown that a subset \(X\) of \(G\) (of finite transcendence degree over \(k)\) extends to a projective subgeometry if and only if the algebraic dependence relations of the elements of \(X\) over a suitable fixed transcendence basis of \(X\) satisfy certain separability conditions. From this, concrete representations of the coordinatizing fields of maximal projective subgeometries of \(G\) as subfields of \(k\) are derived.