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I would like to find a reference to the following statement:

Statement. Let $X$ be a complex projective manifold with an algebraic action of a $k$-dimensional torus $(\mathbb C^*)^k$. Then the cone of curves of $X$ is generated by $(\mathbb C^*)^k$-invariant curves. In particular, each irreducible proper curve in $X$ is linearly equivalent to a sum of such invariant curves with positive coefficients.

(Note, of course, that for toric manifolds - when the action of $(\mathbb C^*)^k$ has an open orbit - this statement is classical.)

PS. I am aware of a proof of this statement - along the lines of the comment below, but would be especially grateful for a precise reference.

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    $\begingroup$ I suppose, given a generator $C$, one can take the flat limit under each $\mathbb{C}^*$ in turn to obtain a new curve $\hat C$ which should be linearly equivalent and torus invariant. $\endgroup$ – Ruadhaí Dervan May 17 '19 at 11:09
  • $\begingroup$ Thank you for this comment! $\endgroup$ – guest170519 May 17 '19 at 17:48
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This follows from a result of Andre Hirschowitz.

Le groupe de Chow equivariant. C. R. Acad. Sci. Paris Ser. I Math. 298 (1984), no. 5, 87--89.

Hirschowitz proved in this paper that for a projective variety with an algebraic action of a connected solvable group $B$, any effective cycle is rationally equivalent to a $B$-invariant effective cycle. The idea is to consider the $B$-action on the Chow variety $Z$ containing a given effective cycle $z$. Applying the Borel fixed point theorem, one finds a $B$-stable cycle $z_0\in \overline{Bz}$.

Of course the Statement follows immediately from Hirschowitz's result.

As it happens, this statement was reproved several times afterwards.

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