TITLE: B-invariant subvarieties
QUESTION [5 upvotes]: Let $B$ be a connected, solvable algebraic group (of dimension 2) acting on a projective variety $Y$ (of dimension $3$): for instance, let $B$ be a Borel subgroup of a reductive algebraic group $G$. 
Question1: Is it always true that there is a $B$-invariant complete flag (i.e. codimension $1$, irreducible and $B$-invariant subvarieties), $Y \supset Y_1\supset Y_2 \supset Y_3$? For my purposes I can assume that there is a point in $Y$ with finite stabliser group and therefore can choose the orbit-closure of that point to be $Y_2$. How about $Y_1$? Is there always a point with $1$-dimensional stabiliser group of the $B$-action on the just described $Y_2$? Obviously there will always exist a $B$-fixed point $Y_3$, by Borel's fixed point theorem, if $Y_1$ is a projective variety on which $B$ acts. 
Question2: Does this generalize to arbitrary dimension $n$ of $Y$ and any reductive algebraic group $G$ with a Borel subgroup $B$?

REPLY [7 votes]: Here is an argument which avoids the use of Hilbert schemes and works for any complete (irreducible) variety $Y$.
As in Jason's answer it suffices to construct a $B$-stable subset $Y_1$ of codimension $1$, provided that $\dim Y\ge1$. For this we consider two cases:

$Y$ does not contain an open $B$-orbit. By Rosenlicht's theorem, there exists an open $B$-stable subset $Y_0\subseteq Y$ such that the orbit space $\pi:Y_0\to Z:=Y_0/B$ exists. Our assumption implies $\dim Z>0$. Therefore, $Z$ contains a subvariety $Z_1$ of codimension $1$. Now take for $Y_1$ the closure of $\pi^{-1}(Z_1)$ in $Y$.
$Y$ does contain an open $B$-orbit $Y_0$. Then the above argument is not applicable since $Z$ is a point. Instead we use the well-known fact that homogeneous spaces for solvable groups are affine. Thus $Y_0$ is affine. But it not complete since $\dim Y_0\ge1$. Thus the complement $C$ of $Y_0$ in $Y$ is non-empty. We conclude with another well-known fact, namely that the complement of an affine open set is always pure of codimension one. So, take for $Y_1$ any component of $C$.