here is my question:

I work over the field of complex numbers, and my schemes are separated and of finite type. Let $X$ be a quasi projective scheme (but not projective), let $Y$ be a regular projective scheme, and let $f: X \to Y$ be a monomorphism of schemes (but not necessary an immersion).

Is the pull-back by $f$ of an ample line bundle on $Y$ an ample line bundle on $X$? And if not, what kind of (not too strong) assumptions I should add on $f$ for this result to be true?

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  • $\begingroup$ What is the definition of ampleness that you are using on general quasi-projective schemes? $\endgroup$ – Francesco Polizzi Jul 14 '14 at 16:26
  • $\begingroup$ I consider the Definition given by Hartshorne, §II.7, where an invertible sheaf $\mathcal{L}$ on a Noetherian scheme $X$ is said to be ample if for every coherent sheaf $\mathcal{F}$ on $X$, there is an integer $n_0 \geq 0$ such that for every $n \geq n_0$, the sheaf $\mathcal{F} \otimes \mathcal{L}^n$ is generated by its global sections. I think it is also the definition considered by Grothendieck in EGA. $\endgroup$ – sabrebooth Jul 15 '14 at 7:53
  • $\begingroup$ Let $X$ be a compact complex manifold and $L$ be an ample line bundle on $X$. Let $Z$ be a set of points in $X$ (taken with multiplicity) and $p:\tilde X→X$ be the blowup of $X$ along $Z$ with exceptional divisor $E$. Then, for $γ$ sufficiently large, $\tilde L=γp^∗L−E$ is ample. $\endgroup$ – user21574 Jul 26 '17 at 4:03

A monomorphism is injective (EGA IV 17.2.6), hence quasi-finite, hence quasi-affine (Zariski main theorem). Now the pull-back of an ample sheaf by a quasi-affine morphism is ample (EGA II 5.1.12).

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