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Definition. A variety $X^r\subset \mathbb{P}^n$ is linearly stable (resp. linearly semistable) if, whenever $L^{n-m-1}\subset \mathbb{P}^n$ is a linear space such that the image cycle $p_L(X)$ of $X$ under the projection $p_L:\mathbb{P}^n-L\longrightarrow \mathbb{P}^m$ has dimension $r$, then $red.deg(p_L(X))>red.deg(X)$ (resp. $red.deg(p_L(X))\ge red.deg(X)$).

This is a definition in [D.Mumford, Stability of projective varieties.]

Mumford give examples about linearly stability in it. But I can't understand...

Why is a genus $0$ curve linearly semistable but not stable?

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I think that Mumford is probably only talking about rational normal curves. By my computation, a generic degree $5$, genus $0$ curve in $\mathbb{P}^3$ does satisfy his definition of linear stability. However, for every rational normal curve $X$ in $\mathbb{P}^n$, the reduced degree equals $1$, i.e. the degree equals $n$. Thus, for a projection that is again a rational normal curve (e.g., projection from a linear space $L$ that has maximal $e_H$), then $\text{red. deg}(p_L(X))$ equals $\text{red. deg}(X)$.

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