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Example 29.38.4. Let $k$ be a field. Consider the graded $k$-algebra

\[ A = k[U, V, Z_1, Z_2, Z_3, \ldots ]/I \quad \text{with} \quad I = (U^2 - Z_1^2, U^4 - Z_2^2, U^6 - Z_3^2, \ldots ) \]

with grading given by $\deg (U) = \deg (V) = \deg (Z_1) = 1$ and $\deg (Z_ d) = d$. Note that $X = \text{Proj}(A)$ is covered by $D_{+}(U)$ and $D_{+}(V)$. Hence the sheaves $\mathcal{O}_ X(n)$ are all invertible and isomorphic to $\mathcal{O}_ X(1)^{\otimes n}$. In particular $\mathcal{O}_ X(1)$ is ample and $f$-ample for the morphism $f : X \to \mathop{\mathrm{Spec}}(k)$. We claim that no power of $\mathcal{O}_ X(1)$ is $f$-relatively very ample. Namely, it is easy to see that $\Gamma (X, \mathcal{O}_ X(n))$ is the degree $n$ summand of the algebra $A$. Hence if $\mathcal{O}_ X(n)$ were very ample, then $X$ would be a closed subscheme of a projective space over $k$ and hence of finite type over $k$. On the other hand $D_{+}(V)$ is the spectrum of $k[t, t_1, t_2, \ldots ]/(t^2 - t_1^2, t^4 - t_2^2, t^6 - t_3^2, \ldots )$ which is not of finite type over $k$.

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