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I'm looking for a reference for the following statement:

Let $G$ be a reductive algebraic group acting on a projective variety $X$. Let $x, y\in X$ be such that $y$ lies in the closure of $Gx$. Then there exists a $1$-dimensional algebraic subtorus $T \subset G$ and a point $x'\in Gx$ so that $y$ lies in the closure of $Tx'$.

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The statement is not correct. Observe that $T$ would be contained in the stabilzer of $y$ but is may very well happen that $G_y$ is unipotent. What you have in mind might be the Hilbert-Mumford criterion, though. In that case $X$ should be affine and the closure of $Tx$ intersects only the closure of the orbit $Gy$ but possibly not the orbit itself.

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