Fujita freeness: how much positivity removes every base point?

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Family 038 · Algebraic and complex geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

A collection of polynomial-like functions can describe a geometric space only where they do not all vanish. If every available section vanishes at the same point, the associated map has a hole there. How much positive twisting guarantees that these base points disappear?

Fujita’s freeness conjecture predicts a remarkably simple answer. On a smooth complex projective variety of dimension $n$, adding $n+1$ copies of any ample line bundle to the canonical bundle should make it globally generated. The threshold depends only on dimension, not on the variety’s equations, degree, or topology.

Family 038 consists of one manuscript, M1, claiming this sharp bound in every dimension. Its subject is freeness: sections should define a morphism everywhere. The stronger requirement that the morphism embed the variety belongs to the separate very-ampleness part of Fujita’s conjecture.

2. Background and essential definitions

Line bundles and maps defined by sections

A line bundle $L$ has a one-dimensional vector space $L_x$ over each point $x$, with compatible local coordinates. A global section is a choice of vector in each fiber which varies algebraically. The finite-dimensional vector space of all global sections is written $H^0(X,L)$.

Suppose $s_0,\ldots,s_N$ form a basis. At a point where they do not all vanish, their values define projective coordinates

$$x\longmapsto[s_0(x):\cdots:s_N(x)].$$

Changing the local trivialization multiplies all these coordinates by the same nonzero scalar, so the projective point is well-defined. A base point is a point where every section vanishes. The line bundle is globally generated, or base-point-free, when there are no base points.

Equivalently, the evaluation map

$$H^0(X,L)\longrightarrow L_x$$

is onto for every $x$. Since the target is one-dimensional, this just asks for at least one section nonzero at each prescribed point. The section can depend on the point. It does not ask for one section which is nonzero everywhere.

Ample and very ample

A line bundle is very ample if its sections define an embedding into projective space. It is ample if some positive tensor power is very ample. An ample line bundle itself need not be globally generated, and it may have no nonzero sections at all.

Global generation is weaker than very ampleness. Sections may define a constant map or identify distinct points. Very ampleness requires enough sections to distinguish points and infinitesimal tangent directions. Confusing these properties would turn the claimed theorem into a stronger assertion.

The canonical bundle $K_X$ consists of top-degree holomorphic differential forms. In local coordinates on an $n$-fold, a form looks like $f\,dz_1\wedge\cdots\wedge dz_n$. An adjoint bundle combines canonical geometry with a positive twist.

M1 uses additive notation:

$$K_X+mL:=K_X\otimes L^{\otimes m}.$$

The plus sign is tensor product, not an addition of spaces of sections. In particular, global generation of one adjoint does not automatically imply global generation after tensoring with an ample bundle which is not itself generated.

Orders of vanishing and singularity centers

A section’s order at a smooth point records the first nonzero term of its local Taylor expansion. More general weighted orders, called valuations, measure vanishing along selected directions or exceptional divisors after blowing up the variety.

A blowup replaces a subvariety by its normal directions. If $E$ is an exceptional divisor on a smooth birational model $\pi:Y\to X$, its log discrepancy is

$$A_X(E)=1+\operatorname{ord}_E(K_Y-\pi^*K_X).$$

It measures the canonical cost of using that divisor to detect vanishing. The center of this order is the subvariety downstairs where it is concentrated.

A log canonical center arises when a weighted effective divisor reaches the limiting singularity strength allowed by discrepancy. In freeness arguments, one tries to create a controlled center at a prescribed point and use a vanishing theorem to extend a nonzero value there to a global section.

3. Development history and previous partial results

Fujita formulated the dimension-dependent adjoint prediction in the development of polarized varieties. A polarized variety is a variety together with a chosen ample line bundle. The conjecture asks for a universal rule applying to every such polarization, rather than a bound tailored to a particular embedding.

For curves, the sharp freeness statement follows from Riemann–Roch. For surfaces, it follows from Reider’s theorem. Ein–Lazarsfeld treated threefolds, Kawamata handled dimension four, and Ye–Zhu treated dimension five. M1’s introduction lists the precise earlier statements and references.

In arbitrary dimension, Angehrn–Siu established a quadratic sufficient bound,

$$m\geq\frac{n^2+n+2}{2}.$$

Subsequent improvements reduced its growth: Helmke developed multiplicity estimates for singular centers, Heier obtained a bound of order $n^{4/3}$, and Ghidelli–Lacini reached order $n\log\log n$.

The recent preprint of Jingjun Han gives the linear bound $m\geq\lceil C_0n\rceil$, with $C_0$ approximately $1.7763$. This is a substantial improvement in the same direction, but it is still larger than $n+1$ in high dimension. The distinction between a linear bound and the sharp linear coefficient is the remaining issue addressed by M1.

M1 also acknowledges Chan’s withdrawn 2024 preprint. The withdrawal notice identifies a local extension step that did not preserve the required vanishing order. That earlier announced solution is therefore not treated here as an established theorem.

Across these approaches, the recurring difficulty is controlling a positive-dimensional singular center while retaining enough positivity to cut it down. A center can be highly singular or have large degree; a universal bound cannot quietly depend on either quantity.

4. What the new manuscripts claim

M1’s Theorem 1.1 states that for every smooth connected projective complex variety $X$ of dimension $n\geq1$ and every ample line bundle $L$, the bundle

$$K_X+(n+1)L$$

is generated by global sections. Corollary 6.3 extends this to every integer $m\geq n+1$.

There is no assumption that $L$ itself is generated or effective. Connectedness serves only to simplify notation; the same assertion can be applied separately to connected components. In dimension zero, every line bundle is globally generated.

Why the threshold cannot be lowered

On projective space, let $X=\mathbb P^n$ and $L=\mathcal O_{\mathbb P^n}(1)$. Its canonical bundle is $\mathcal O_{\mathbb P^n}(-n-1)$, so

$$K_X+(n+1)L\simeq\mathcal O_X, \qquad K_X+nL\simeq\mathcal O_{\mathbb P^n}(-1).$$

The first has a constant nonzero section and is generated. The second has no nonzero global sections. Thus no theorem with $n$ replacing $n+1$ can hold uniformly for all smooth polarized varieties.

This example also emphasizes the difference from very ampleness. The generated trivial bundle defines a constant map, not an embedding of positive-dimensional projective space.

Why all larger exponents follow

Because $L$ need not be generated, simply tensoring by extra copies does not justify the corollary. M1 instead applies the sharp statement to a product with projective space. For $r=m-n-1\geq0$, use

$$X\times\mathbb P^r,\qquad L\boxtimes\mathcal O_{\mathbb P^r}(1),$$

where $\boxtimes$ means the tensor product of the two pulled-back bundles. The resulting adjoint is

$$(K_X+mL)\boxtimes\mathcal O_{\mathbb P^r}(n).$$

Restricting a generated bundle to $X\times\{z\}$ gives generation of $K_X+mL$. The product argument retains the arbitrary-ample hypothesis.

5. Why the advance matters

Importance: 5/5, provisional. Fujita freeness is a central effective-positivity conjecture with a sharp and easily stated threshold. Its claimed resolution would replace bounds depending on more geometry, or larger functions of dimension, by the optimal universal number of ample twists.

The practical mathematical gain is control of an actual map. Ampleness already promises that sufficiently large powers eventually behave well. The conjecture says exactly how the canonical correction makes that behavior uniform across all smooth projective complex varieties.

The sharpness example shows that the constant is structural, rather than an artifact of a particular proof. The score concerns the significance of the stated theorem if correct and does not include the still stronger very-ampleness conclusion.

6. The main idea

Convert a base point into excessive vanishing

Fix a point $x$ and suppose that every section of $K_X+(n+1)L$ vanishes there. The manuscript first translates this failure into a strict asymptotic statement about sections of high powers $qL$: bases adapted to ordinary vanishing at $x$ have normalized mean order exceeding $n/(n+1)$.

This moves the problem to a large finite-dimensional space where sections can be counted. The goal is to show that the resulting excess cannot coexist with the geometric constraints on their vanishing orders.

Minimize a quantity that tracks discrepancy and section orders

The paper considers bases $(s_i)$ of $H^0(X,qL)$ and weighted monomial orders $v$ whose closed centers contain $x$. For a parameter $t<n+1$, the quantity is

$$\frac{1}{h^0(X,qL)} \sum_i\exp\left(\frac{A_X(v)}{t}-\frac{v(s_i)}{q}\right).$$

The discrepancy term charges for the singularity detector; the section-order term rewards simultaneous vanishing. Varying the scale and tangential directions gives different inequalities from the same objective.

An infimum alone would be insufficient: the argument needs an attained minimizer. M1 encodes simultaneous order conditions on a parameter space of section flags and uses threshold semicontinuity and a fixed resolution to obtain one. It does not assume that all possible bases or all possible birational models form a compact space.

Rule out a positive-dimensional center

Suppose a minimizing order were centered on a positive-dimensional subvariety through $x$. Tangential variation gives an upper bound on a weighted moment of leading Taylor exponents. A restriction-space estimate and the Brunn–Minkowski inequality give a conflicting lower bound.

Brunn–Minkowski is a convex-geometric principle controlling how volume grows under addition of sets. Here the sets organize possible leading exponents of sections. The manuscript’s required uniformity is over all centers of a fixed dimension, including centers with unbounded degree or singularity complexity.

Eliminating those centers leaves an order concentrated at the prescribed point. This is the geometric reduction needed for the final extension step.

Extend one nonzero value

A rational perturbation isolates a reduced divisor above $x$ where the discrepancy equality occurs. Kawamata–Viehweg vanishing then yields surjectivity of a restriction map on the resolved space. A nonzero value in the one-dimensional fiber over $x$ can be lifted and descended to a global section nonzero at $x$.

This contradicts the assumed base point. The manuscript controls the needed singularities near the equality locus; it does not assume a globally log canonical boundary at that stage. Since the starting point was arbitrary, removing every possible base point proves the claimed generation statement.

7. Scope, limitations, and open questions

The theorem concerns smooth projective varieties over $\mathbb C$ and ample line bundles. It is not stated for arbitrary singular varieties, positive characteristic, or arbitrary nef bundles. Ampleness supplies quantitative section growth used throughout the argument.

Global generation does not imply an embedding. Fujita’s very-ampleness prediction concerns separation of points and tangent directions at the next threshold, and is not claimed here. Stronger formulations using prescribed intersection numbers rather than one ample line also require their own statements.

The bound is uniform and sharp, but the manuscript is not an algorithm for computing a basis of sections from arbitrary defining equations. The claimed existence of a nonzero section at each point is a geometric result, not a stated computational complexity bound.

The historical withdrawal and earlier quantitative theorems are reported with attribution. The present exposition takes M1’s claim as stated and does not independently verify its proof.

8. Sources and further reading

M1 §1 states the theorem and reviews the preceding bounds. Section 3 turns a base point into an order inequality, §4 establishes attainment, §5 excludes positive-dimensional centers, and §6 constructs the final section. Corollary 6.3 contains the product argument for all larger exponents.

The linked Reider, Ein–Lazarsfeld, Angehrn–Siu, Ghidelli–Lacini, and Han papers describe important stages of the development. For optional background, see ample line bundles and linear systems of divisors.

Source manuscripts

• M1. OpenAI, Fujita’s freeness conjecture (September 23, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.