Lemma 29.41.7. A composition of H-projective morphisms is H-projective.

**Proof.**
Suppose $X \to Y$ and $Y \to Z$ are H-projective. Then there exist closed immersions $X \to \mathbf{P}^ n_ Y$ over $Y$, and $Y \to \mathbf{P}^ m_ Z$ over $Z$. Consider the following diagram

Here the rightmost top horizontal arrow is the Segre embedding, see Constructions, Lemma 27.13.6. The diagram identifies $X$ as a closed subscheme of $\mathbf{P}^{nm + n + m}_ Z$ as desired. $\square$

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