Configurations of Linear Subspaces and Rational Invariants

摘要

Let Grn.d(C) denote the Grassmannian of all d-dimensional linear subspaces in (Cn, and let GLn(C) x (Grn.d(C))s ->(Grn.d(C))s be the canonical diagonal action. Dolgachey [DB] posed the following question: Is the quoiient Grn,2(C)s/GLn(C) (e.g., in the sense of Rosenlicht) alwavs rational? Recall that a Rosenlicht quotient of an algebraic variety X acted on by an al- gebraic group G is an algebraic variety V together with a rational map X -> V whose generic fibers coincide with the G-orbits. Such quotients always exist and are unique up to birational isomorphisms [R] . In the sequel all quotients will be as- sumed of this type. An algebraic variety Q is rational if it is birationally equivalent to Pm with m = dim Q.