  Let $G$ be a complex connected reductive algebraic group that acts on a
smooth complex algebraic variety $X$, and let $E$ be a $G$-equivariant
algebraic vector bundle over $X$. A section of $E$ is regular if it is
transversal to the zero section. Let $U\subset\Gamma(X,E)$ be the subset of
regular sections. We give a sufficient condition in terms of topological
invariants of $E$ and $X$ that implies that every orbit map $O\colon G\to U$
induces a surjection in rational cohomology. Under natural assumptions on $X$
and $E$ this condition is also necessary.
  If the condition is satisfied, then (1) the geometric quotient $U/G$ exists;
(2) there is an isomorphism $H^*(U,\mathrm{Q})\cong H^*(G,\mathrm{Q})\otimes
H^*(U/G,\mathrm{Q})$ of cohomology rings; (3) the order of the stabiliser
$G_s,s\in U$ divides a certain expression that can be explicitly calculated
e.g. if $X$ is a compact homogeneous space. In some cases (e.g. if $E$ is a
line bundle) we also prove similar statements for the space of the zero loci of
$s\in U$. We apply these results to several explicit examples which include
hypersurfaces in projective spaces, non-degenerate quadrics and complete flag
varieties of the simple Lie groups of rank 2, and also certain Fano varieties
of dimension 3 and 4.
