Does $\mathscr{L}^{\otimes n}$ very ample imply $\mathscr{L}^{\otimes (n+1)}$ very ample? (EGA II, Remark 4.5.12)

EGA II (Grothendieck, 1961) leaves open, in Remark 4.5.12, whether an invertible sheaf (line bundle) $\mathscr{L}$ on a scheme $X$ whose $n$-th tensor power is very ample relative to a morphism $q : X \to Y$ must also have $\mathscr{L}^{\otimes(n+1)}$ very ample. The set of very ample multiples $S(\mathscr{L})$ is known to be closed under addition (via the Segre embedding) and, for $q$ of finite type over an affine base, to contain all sufficiently large integers; the question is whether $S(\mathscr{L})$ is an upward-closed interval $\{m \ge n_0\}$, so that a single threshold governs when powers of $\mathscr{L}$ define projective embeddings. The obstruction is sharp: the standard criterion — a very ample sheaf tensored with a globally generated sheaf is very ample — does not apply, since an ample sheaf with a very ample power need not itself be globally generated. Sixty-five years later the question remains open: no proof is known even for smooth projective varieties over an algebraically closed field, and no counterexample is known in any setting within the finite-type hypotheses; an unanswered MathOverflow thread from June 2026 on the associated numerical semigroup problem is consistent with its status as open among experts.

Background

An invertible sheaf (line bundle) $\mathscr{L}$ on a scheme $X$ is a sheaf of $\mathscr{O}_X$-modules that is locally free of rank one; its tensor powers $\mathscr{L}^{\otimes n}$ play the role of multiples of a divisor. Given a morphism $q : X \to Y$, the sheaf $\mathscr{L}$ is called very ample relative to $q$ (EGA II, definition 4.4.2) if there exist a quasi-coherent $\mathscr{O}_Y$-module $\mathscr{E}$ and a surjection $q^*\mathscr{E} \to \mathscr{L}$ such that the induced morphism $X \to \mathbf{P}(\mathscr{E}) = \operatorname{Proj}_Y \operatorname{Sym}(\mathscr{E})$ is an immersion (a composite of a closed and an open immersion) and $\mathscr{L}$ is the pullback of the tautological sheaf $\mathscr{O}(1)$ (the universal quotient line bundle on the projective bundle). In the absolute case $Y = \operatorname{Spec} k$ with $q$ of finite type this recovers the classical notion: the complete linear system of $\mathscr{L}$ embeds $X$ as a locally closed subscheme of a projective space over the field $k$. The Stacks Project (Lemma 29.39.7) records the equivalent forms of this definition used in the modern literature, notably the form via the canonical map $q^* q_* \mathscr{L} \to \mathscr{L}$. The sheaf $\mathscr{L}$ is ample relative to $q$ (EGA II, 4.5) if the non-vanishing loci of sections of the powers $\mathscr{L}^{\otimes n}$ generate the topology of $X$, equivalently if the canonical morphism from $X$ to the homogeneous spectrum of the section ring $\bigoplus_{m \ge 0} H^0(X, \mathscr{L}^{\otimes m})$ (the graded ring of sections of all powers) is an open immersion (EGA II, Theoreme 4.5.2). Ampleness is the fundamental positivity condition in algebraic geometry: it underlies projective embeddings, Serre's cohomological vanishing criteria, and the construction of homogeneous coordinate rings.

Two classical facts frame the question studied here. First (EGA II, Proposition 4.4.8, proved with the Segre morphism, that is, the closed immersion $\mathbf{P}(\mathscr{E}) \times_Y \mathbf{P}(\mathscr{F}) \hookrightarrow \mathbf{P}(\mathscr{E} \otimes \mathscr{F})$ that realizes tensor products of very ample sheaves as very ample): if $\mathscr{M}$ is very ample relative to $q$ and $\mathscr{N}$ is an invertible sheaf admitting a surjection $q^*\mathscr{F} \to \mathscr{N}$ from the pullback of a quasi-coherent $\mathscr{O}_Y$-module $\mathscr{F}$ (over an affine base with $q$ quasi-separated, it suffices that $\mathscr{N}$ be generated by its global sections), then $\mathscr{M} \otimes \mathscr{N}$ is very ample relative to $q$. Second (EGA II, Proposition 4.5.10): if $Y$ is affine and $q$ is separated (its diagonal $X \to X \times_Y X$ is a closed immersion) and of finite type, then $\mathscr{L}$ is ample if and only if $\mathscr{L}^{\otimes n}$ is very ample relative to $q$ for all sufficiently large $n$. Consequently, for such $q$ and ample $\mathscr{L}$, the set $S(\mathscr{L}) = \{ m \ge 1 : \mathscr{L}^{\otimes m} \text{ is very ample relative to } q \}$ is cofinite in the positive integers, and it is closed under addition ($a, b \in S$ implies $a + b \in S$, by the first fact since a very ample sheaf is generated from the base); thus $S(\mathscr{L}) \cup \{0\}$ is a numerical semigroup (a cofinite additive submonoid of $\mathbb{N}$).

EGA II, Remark 4.5.12 — the subject of this question — asks whether this semigroup is as simple as possible: an interval $\{m \ge n_0\}$. Concretely: if $\mathscr{L}^{\otimes n}$ is very ample relative to $q$, must $\mathscr{L}^{\otimes(n+1)}$ be? The difficulty is structural. Writing $\mathscr{L}^{\otimes(n+1)} = \mathscr{L}^{\otimes n} \otimes \mathscr{L}$, the first factor is very ample and the second is ample (a power of it is very ample), but an ample invertible sheaf need not be globally generated, so the standard criterion does not apply, and no replacement criterion for very ampleness of such tensor products is known. All well-understood special cases exhibit interval behavior: if $\mathscr{L}$ itself is very ample then every power is (Segre); on abelian varieties (smooth projective varieties carrying the structure of an algebraic group) a Lefschetz-type theorem gives $\mathscr{L}^{\otimes n}$ very ample for all $n \ge 3$ whenever $\mathscr{L}$ is ample, so $S(\mathscr{L})$ is one of $\mathbb{N}$, $\mathbb{N}_{>1}$, $\mathbb{N}_{>2}$; and on smooth curves the classical degree bound (a line bundle of degree at least $2g+1$, where $g$ is the genus, the dimension of the space of regular differential forms on the curve, is very ample) again forces interval behavior. At the boundary of the finite-type hypotheses the picture degenerates: the Stacks Project (Example 29.39.4) exhibits a scheme $X = \operatorname{Proj} k[U, V, Z_1, Z_2, \ldots]/(U^2 - Z_1^2, U^4 - Z_2^2, U^6 - Z_3^2, \ldots)$ ($\operatorname{Proj}$ denoting the scheme of homogeneous prime ideals of a graded ring), not of finite type over $k$, on which $\mathscr{O}(1)$ is ample yet no power is very ample. The open question therefore lives naturally in the finite-type setting, where it appears to remain unresolved in the published literature.

Problem Statement

Let $Y$ be an affine scheme and let $q : X \to Y$ be a separated quasi-compact morphism — the standing hypotheses of EGA II, §4.5, where the question below is posed; here “separated” means that the diagonal $X \to X \times_Y X$ is a closed immersion, and “quasi-compact” that inverse images of affine opens are covered by finitely many affine opens. The results on powers of an ample sheaf that frame the question, Proposition 4.5.10 and Corollary 4.5.11, assume in addition that $q$ is of finite type, and that finite-type case is the motivating instance. Let $\mathscr{L}$ be an invertible $\mathscr{O}_X$-module. Say $\mathscr{L}$ is very ample relative to $q$ if there exist a quasi-coherent $\mathscr{O}_Y$-module $\mathscr{E}$ and a surjection $q^*\mathscr{E} \to \mathscr{L}$ such that the induced morphism $X \to \mathbf{P}(\mathscr{E})$ is an immersion and $\mathscr{L}$ is the pullback of $\mathscr{O}(1)$ (EGA II, 4.4.2; equivalent forms in the Stacks Project, Lemma 29.39.7). Define the set of very ample multiples

$S(\mathscr{L}) = \{ m \ge 1 : \mathscr{L}^{\otimes m} \text{ is very ample relative to } q \}.$

Question (EGA II, Remark 4.5.12). Suppose $S(\mathscr{L}) \neq \varnothing$, i.e. $\mathscr{L}^{\otimes n}$ is very ample relative to $q$ for some $n \ge 1$. Does it follow that $\mathscr{L}^{\otimes(n+1)}$ is very ample relative to $q$? Equivalently: is $S(\mathscr{L})$ upward closed — of the form $\{m \ge n_0\}$ — so that, in the finite-type case where $S(\mathscr{L})$ is a cofinite additive semigroup, it is a pure interval and a single threshold governs very ampleness of the powers?

The special case $Y = \operatorname{Spec} k$ for an algebraically closed field $k$ with $X$ a smooth projective variety over $k$ is open; it is the most informative instance of the question.

References

1. Alexander Grothendieck, Éléments de géométrie algébrique II: Étude globale élémentaire de quelques classes de morphismes, Publications Mathématiques de l'IHÉS 8 (1961), 5-222. DOI: 10.1007/BF02699291. Full text: http://www.numdam.org/item/PMIHES_1961__8__5_0/
2. Holger Brenner and Stefan Schröer, Ample families, multihomogeneous spectra, and algebraization of formal schemes, Pacific Journal of Mathematics 208 (2003), no. 2, 209-230. DOI: 10.2140/pjm.2003.208.209. arXiv: math/0012082, https://arxiv.org/abs/math/0012082
3. Dennis S. Keeler, Ample filters of invertible sheaves, Journal of Algebra 259 (2003), 243-283. DOI: 10.1016/s0021-8693(02)00557-4. arXiv: math/0108068, https://arxiv.org/abs/math/0108068
4. The Stacks Project Authors, The Stacks Project, Example 29.39.4 (01VO) and Lemma 29.39.7 (01VR), chapter Morphisms of Schemes, section Very ample sheaves. https://stacks.math.columbia.edu/tag/01VO and https://stacks.math.columbia.edu/tag/01VR
5. Tuong Le, Properties of the numerical semigroup of very ample multiples of an ample divisor?, MathOverflow question 512531 (22 June 2026, unanswered; comment by user cll on the abelian variety case). https://mathoverflow.net/questions/512531/properties-of-the-numerical-semigroup-of-very-ample-multiples-of-an-ample-diviso
6. asv (pseudonymous poster) with a comment by R. van Dobben de Bruyn, Two definitions of very ample line bundle and a property its tensor powers, MathOverflow question 447277 (22 May 2023, unanswered). https://mathoverflow.net/questions/447277/two-definitions-of-very-ample-line-bundle-and-a-property-its-tensor-powers