Example 29.38.5. Let $k$ be an infinite field. Let $\lambda _1, \lambda _2, \lambda _3, \ldots $ be pairwise distinct elements of $k^*$. (This is not strictly necessary, and in fact the example works perfectly well even if all $\lambda _ i$ are equal to $1$.) Consider the graded $k$-algebra
\[ A_ d = k[U, V, Z]/I_ d \quad \text{with} \quad I_ d = (Z^2 - \prod \nolimits _{i = 1}^{2d} (U - \lambda _ i V)). \]
with grading given by $\deg (U) = \deg (V) = 1$ and $\deg (Z) = d$. Then $X_ d = \text{Proj}(A_ d)$ has ample invertible sheaf $\mathcal{O}_{X_ d}(1)$. We claim that if $\mathcal{O}_{X_ d}(n)$ is very ample, then $n \geq d$. The reason for this is that $Z$ has degree $d$, and hence $\Gamma (X_ d, \mathcal{O}_{X_ d}(n)) = k[U, V]_ n$ for $n < d$. Details omitted.
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