The Stacks project

Lemma 53.3.3. Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. If $X$ has a $\mathfrak g^ r_ d$, then $X$ has a $\mathfrak g^ s_ d$ for all $0 \leq s \leq r$.

Proof. This is true because a vector space $V$ of dimension $r + 1$ over $k$ has a linear subspace of dimension $s + 1$ for all $0 \leq s \leq r$. $\square$

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