Fujita’s freeness conjecture

1 Introduction

We use additive notation for line bundles. Fujita’s freeness conjecture [Fujita(1987)] predicts that the canonical bundle becomes globally generated after twisting by n+1 copies of an arbitrary ample line bundle on a smooth projective n-fold. The ample bundle itself need not have any nonzero sections. We prove the following precise form.

The connectedness assumption only fixes notation: the assertion applies to each connected component. In dimension zero every line bundle is globally generated.

The exponent is sharp: for X=\mathbb P^n and L=\cO_{\mathbb P^n}(1), the bundle K_X+(n+1)L is trivial, whereas K_X+nL has no nonzero sections. Global generation asks for a nonvanishing section at each point; it does not assert separation of points or tangent directions, or the very-ampleness part of Fujita’s conjecture. Theorem 1.1 also implies that K_X+mL is globally generated for every integer m\geq n+1, even when L is not globally generated; see Corollary 6.3.

History and methods

For curves, Riemann–Roch gives the sharp bound. The surface case follows from Reider’s theorem, and the threefold, fourfold, and fivefold cases were proved by Ein–Lazarsfeld, Kawamata, and Ye–Zhu, respectively [Reider(1988), Ein and Lazarsfeld(1993), Kawamata(1997), Ye and Zhu(2020)]. In arbitrary dimension, Angehrn–Siu obtained generation for m\geq (n^2+n+2)/2 [Angehrn and Siu(1995)]. Helmke developed multiplicity estimates for the centers appearing in the vanishing argument [Helmke(1997), Helmke(1999)]; Heier combined these estimates with the Angehrn–Siu method to obtain a bound of order n^{4/3} [Heier(2002)]. Ghidelli–Lacini subsequently obtained the bound m\geq n(\log\log n+2.34) for n\geq2 [Ghidelli and Lacini(2024), Theorem 1.1]. Han’s recent preprint gives m\geq\lceil C_0n\rceil, where C_0=1.77629988\ldots [Han(2026), Theorem 1.1]. Chan announced a proof of the stronger intersection-number formulation in [Chan(2024), Theorem 1.3]; that preprint was withdrawn after the local extension argument did not guarantee the required vanishing order.

The common vanishing strategy constructs controlled singularities at the prescribed point, reduces a suitable center to a point, and then lifts a nonzero fiber value [Ein and Lazarsfeld(1993), Kawamata(1997)]. A recurring difficulty is that a positive-dimensional center can itself be singular, and cutting it down costs positivity. Angehrn–Siu used movement to a smooth point and semicontinuity. Fujita estimated ranks of restriction maps to the exceptional divisor of a point blowup, while Helmke bounded the multiplicities of log canonical centers [Angehrn and Siu(1995), Fujita(1993), Helmke(1997), Helmke(1999)]. Our argument controls a minimizing center by comparing the growth of images of restriction maps with a tangential first variation. The estimate is uniform over all integral subvarieties of a fixed dimension, including singular subvarieties of unbounded degree. The final lift uses the local discrepancy-and-vanishing mechanism of [Fujita(1993), §1(1.6)]; the minimizing construction produces the configuration to which it applies.

The convex viewpoint has a related ancestry. Flag valuations, initial-exponent semigroups, and their relation to the dimensions of section spaces appear in Okounkov’s work [Okounkov(2003)] and are developed systematically by Lazarsfeld–Mustaţă and Kaveh–Khovanskii [Lazarsfeld and Mustaţă(2009), Kaveh and Khovanskii(2012)]. Boucksom–Chen study the asymptotic distributions of filtration jumps [Boucksom and Chen(2011)], and Blum–Jonsson prove attainment for valuative log-canonical and stability thresholds of ample bundles [Blum and Jonsson(2020), Theorem E]. The exponential average considered here is a different finite-degree objective, whose attainment is proved in Section 4. The center argument uses elementary counts of initial exponents and the Brunn–Minkowski inequality directly.

The proof

Section 2 fixes the discrepancy normalization and the threshold and vanishing results used below. Fix a point x and suppose that the adjoint bundle is not generated there. An equality-case argument first gives a strict asymptotic inequality for bases of H^0(X,mL) adapted to ordinary vanishing at x: their mean order divided by m has lower limit greater than n/(n+1) (Section 3). We then minimize

over bases (s_i) of H^0(X,mL) and weighted monomial orders v whose closed centers contain x. Here \A(v) is the log discrepancy, with the same scaling as v. This functional links three parts of the proof: its derivative under scaling detects the strict ordinary-order excess; its derivative in tangential directions measures a weighted first moment; and its gradient gives positive coefficients for the rational divisor used in the final lift.

Two obstacles enter this approach. First, both bases and models vary. Simultaneous lower bounds on normalized section orders can be encoded by an ideal on the product of X and the complete flag variety of H^0(X,mL). Threshold semicontinuity and retraction to one log resolution then give attainment of the minimum, without compactifying the space of all bases or all valuations (Section 4). Second, a first variation in directions tangent to a positive-dimensional center gives an upper bound for a weighted moment of initial exponents. A uniform restriction-space estimate and Brunn–Minkowski give a contradictory lower bound. The uniformity is independent of the degree and singularities of the moving center (Section 5).

The resulting point-centered minimizer determines a supporting functional on a finite collection of rational polytopes of normalized section orders. A small ideal perturbation makes discrepancy equality rigid along a reduced divisor S over x. Kawamata–Viehweg vanishing then lifts a nonzero fiber value at x. Only local control near that equality locus is needed; the construction does not assume a globally log canonical boundary (Section 6). Figure 1 records the quantities passed between these steps.

2 Orders, thresholds, and vanishing

Throughout the proof, X, L, and n satisfy the hypotheses of Theorem 1.1. Write

Ampleness gives

All centers are closed centers. Thus a divisor centered along a subvariety W is an admissible test at every point of W, not just at its generic point.

2.1 Weighted orders

We use resolution and principalization in characteristic zero with a prescribed simple normal crossing (SNC) boundary; see [Włodarczyk(2005), Theorems 1.0.1 and 2.4.1]. Models used for vanishing are projective. We may resolve above an already chosen model and retain any previously extracted prime by its strict transform.

For a prime divisor F on a smooth birational model \pi:Y\to X, the log discrepancy is

We permit positive multiples of \ord_F and scale discrepancy by the same multiple. Orders of sections mean orders of their effective zero divisors; orders of ideals mean the minimum of the orders of local generators. A local frame pulled back from X may be used in these calculations.

We also use weighted monomial orders. Choose a smooth projective birational model and local regular coordinates adapted to the entire relative canonical divisor: every component of that divisor meeting the coordinate neighborhood is a coordinate hyperplane. All coordinate divisors used below, together with that support, must form a simple normal crossing divisor. Choose a subset E_1,\ldots,E_r of the coordinate divisors and give it positive real weights u_1,\ldots,u_r; the remaining coordinates have weight zero. The order of a germ is the least weight of a nonzero monomial in their local equations, with coefficients in the remaining coordinates, and

A coordinate divisor absent from the relative canonical divisor has log discrepancy one. We use a chosen irreducible component of the positive-support stratum, whose generic point avoids all zero-weight coordinate divisors; its closed image is the center on X. This adaptation condition is essential for (2.2). The test with all weights zero is excluded.

For positive rational weights this is a divisorial order up to scale. Indeed, clearing denominators and extracting the corresponding primitive ray is a toroidal modification in regular parameters. The relative canonical order in those parameters is the sum of the primitive weights minus one. Adding the pullback of the previous relative canonical divisor gives (2.2). The extraction can be retained on a smooth projective model by resolution. Approximating weights on a fixed support by positive rational weights approximates both discrepancy and the orders of any fixed finite collection of sections. For the latter assertion only finitely many coordinatewise minimal exponents need be considered.

2.2 Thresholds in families

For a coherent ideal \mathfrak a on a smooth variety, its local log canonical threshold at x is computed on a log resolution by

The minimum is +\infty for the unit ideal. For the zero ideal the threshold is zero. The same formula follows by change of variables from local integrability of (\sum |g_i|^2)^{-c} for generators g_i of \mathfrak a.

2.3 Vanishing and descent

The cohomological input is Kawamata–Viehweg vanishing [Kawamata(1982), Viehweg(1982)], in the following rounding formulation [Fujino(2009), Theorem 0.1].

This is the rounding formulation with Q=D-K_T-\Delta: its fractional support is SNC and K_T+\ceil{Q}=D. In particular, for an integral divisor M and an effective rational SNC divisor B, nefness and bigness of M-B give vanishing for K_T+M-\floor{B}, by taking \Delta=\{B\}.

We will repeatedly use the following elementary descent observation.

3 The ordinary-order alternative

This section converts failure of adjoint generation at a point into a strict asymptotic inequality for ordinary orders. Lattice counting gives the non-strict bound; the main task is to show that its equality case already permits a nonzero fiber value to be lifted.

Fix a closed point x\in X. Choose regular parameters at x and a local frame of L, using its powers as frames of jL. We order Taylor monomials first by total degree and then lexicographically, always taking the smallest monomial. Let

be the set of initial exponents. Thus |\operatorname{in}(s)|_1=\mult_x(s).

The equality between the number of initial exponents and the dimension of the section space is the elementary counting principle in [Lazarsfeld and Mustaţă(2009), Lemma 1.4] and [Kaveh and Khovanskii(2012), Proposition 2.6]. We include its proof for the order chosen here.

4 Finite-degree exponential minimization

We now allow both the basis and the monomial test to vary. The strict ordinary-order inequality makes the objective less than one, and a supply of low-order jets bounds the discrepancy on that sublevel set. To prove attainment, we first select one flag and its adapted basis; only then do we choose the resolution on which compactness is used.

For N_m>0, an ordered basis \mathbf s=(s_1,\ldots,s_{N_m}) of R_m, a nonzero monomial test v, and t>0, put

Here c_X(v) denotes the closed center, and the infimum also ranges over the bases. Positive rescaling of a test is always allowed. Approximating its positive weights by rational weights preserves its center and approximates the orders of finitely many sections and its discrepancy. Thus the same infimum is obtained using positive multiples of prime-divisor orders.

The same closed realizability loci also compare the infimum along a possible center. This comparison will allow the tangential variation in Section 5 to be tested at a very general point.

5 Eliminating positive-dimensional centers

The specialization comparison can be combined with a tangential variation of a minimizing test. The essential point is that the estimates for sections on its center are uniform as that center varies.

Here a minimizing pair consists of a monomial test and a basis attaining the infimum. The assertion concerns all such pairs, even if their models, coordinates, and bases differ. This universal assertion will be needed in Section 6: a second test with the same normalized section orders and discrepancy is also minimizing, so it too must have point center. We first establish a uniform estimate that will apply to all possible centers.

5.1 A uniform estimate on restriction spaces

Fix a positive integer k such that kL is very ample, and use its complete system to embed X in a projective space. For an integral d-dimensional closed subvariety W\subset X, put

We require a lower estimate for this image, rather than surjectivity of the restriction map.

5.2 A minimizing test and its tangent directions

Assume from now on that P is not generated at x. Fix t\in[t_0,n+1] and a sufficiently large degree m in the ranges of Proposition 4.4, and write f_j=f_{j,t}(x). Suppose that a minimizing monomial pair has test v and center W of dimension d>0. We derive upper and lower bounds for its weighted tangential moment, keeping m and t fixed in this calculation. Let Z be the closure of its defining stratum on a smooth model Y\to X, and let r be the number of its positive weights. Put k'=n-d, so r\le k'. For the local estimates below, keep this test, its model, and the coordinates chosen below fixed; section spaces in every degree j use the same Taylor order.

Choose a very general smooth point y\in W satisfying Lemma 4.5 in degree m:

The very general condition and the smooth open sets used below can be met simultaneously over \C. Above y choose a point z of the open stratum of Z where Z\to W is smooth, avoiding all additional components of the relative canonical divisor. Generic smoothness allows these choices. Take regular parameters at z in three groups:

1. 1.

   the r parameters defining the positive-weight crossing components;

2. 2.

   k'-r parameters along the fiber of Z\to W;

3. 3.

   d parameters pulled back from affine linear projective coordinates on W vanishing at y.

For the last group, choose d independent differentials from the fixed kL embedding. Complete them by the fiber parameters using the smoothness of Z\to W.

The point z lies in the open positive-support stratum, the positive equations change only by units, and the complementary parameters have weight zero. By Lemma 2.1, the original weighted order is therefore unchanged and can be computed from Taylor series at z.

Choose a local frame of L near y and pull it back to Y. The terms of a section with all first r exponents zero are its restriction to Z. Since that restriction comes from W, its Taylor expansion depends only on the last d parameters.

Write Taylor exponents as

and let \ell(p) be their old weighted order; the middle parameters have weight zero. Order monomials first by increasing \ell(p), then by increasing |\beta|_1, and finally by a fixed additive monomial well-order, such as total degree followed by lexicographic order. This combined order is a well-order: below any fixed weighted bound there are only finitely many exponent choices in the positive-weight variables; the tangential total degree and the last tie-break then have least elements. It is also additive.

Let \Gamma_j be the set of possible initial exponents of nonzero sections of R_j in these coordinates. The leading coefficient has one-dimensional leaves, so elimination gives

The cardinality follows by the elimination argument in the proof of Lemma 3.1; compare the one-dimensional-leaf count in [Kaveh and Khovanskii(2012), Proposition 2.6]. Multiplication adds initials because the order is additive and the local frames in all degrees are powers of the same frame.

Take a degree-m basis representing the distinct elements of \Gamma_m. Since the initial order first compares v-values, this basis is adapted to the v-filtration: its sorted valuation values dominate those of any basis. Replacing the original minimizing basis by this one cannot increase the objective, so the new pair still realizes the global minimum. For this fixed test and reference degree m, put

In particular,

We first use minimality to bound the weighted mean tangential degree in \Gamma_m from above.

Perturb v at z by giving weight \epsilon to each of the d tangential parameters and weight \epsilon^2 to each of the k'-r fiber parameters, while leaving its positive weights unchanged. For \epsilon>0 this is an allowed monomial test v_\epsilon with center z on the model and center y on X. Since the added coordinate divisors have discrepancy one near z,

For a section of the chosen basis with initial exponent (p,\beta),

To justify this for Taylor series, fix the section. There are only finitely many positive-coordinate exponent choices below any fixed old weighted bound, so there is a positive gap from its least old weight to the next larger weight in a bounded interval. Nonnegativity of every exponent then excludes higher old weights from the minimum for sufficiently small \epsilon. Among terms of least old weight, the least tangential total degree determines the coefficient of \epsilon; its value is |\beta|_1. Remaining differences affect only the \epsilon^2 coefficient. One term realizing both least quantities provides a finite upper bound for that coefficient. The claim follows. There is no need to choose \epsilon uniformly as m varies: the differentiation is performed for one fixed finite basis at a time.

Let F(\epsilon) be the objective of that basis and v_\epsilon in degree m. By (5.4),

Its right derivative at zero is therefore nonnegative. Differentiating using (5.8) and (5.9), and canceling positive common factors, gives

5.3 Growth forced by restricted sections

We now bound the same moment T_m from below. Restrictions to W give many pure tangential initials; multiplication by their sections forces every nonempty transverse slice to grow with the degree. For a fixed transverse exponent p, define

with b_p(j)=0 for an empty slice. The restriction observation above implies

Indeed a nonzero restriction has old weight zero and no fiber variables; elimination within the restriction space gives distinct pure tangential initials. Lemma 5.2 consequently gives

for all sufficiently large h, uniformly in the center and the test.

There is also a useful exact inclusion in bounded degree. Set h_0=kd. Write the selected affine projective coordinates as s_i/s_0, with s_0(y)\ne0 and s_i(y)=0, for 1\le i\le d. For \beta\in\{0,1\}^d, the section

has initial exponent (0,\beta). The change from the projective trivialization to our local frame is a unit, whose initial exponent is zero. Hence

The cube inclusion turns the continuous Brunn–Minkowski inequality into the discrete estimate needed here. If A,C are nonempty finite subsets of \mathbb Z^d, then

The volume of a finite lattice set thickened by the unit cube is its cardinality, since different cubes have disjoint interiors. The Brunn–Minkowski inequality [Gardner(2002), Theorem 4.1], applied to these finite unions of cubes, gives

Apply this with A=G_p(j) and C=G_0(h-h_0). Equations (5.5) and (5.12) show that their sum with the cube lies in G_p(j+h). Thus, whenever G_p(j) is nonempty and h is sufficiently large,

The constants and degree threshold remain independent of j, p, the center, and its chosen coordinates.

To sum these slice-growth estimates with the exponential weights, put

using the same fixed test and reference degree m as in (5.6). All these sums are finite. For sufficiently large h, we have h-C'_d\ge0. Convexity of the (d+1)st power and (5.14) yield

Only slices present at degree j are needed in deriving this estimate; new slices make a nonnegative contribution. The coefficient Q_j appears because b_p(j)^d/d!=\#G_p(j).

The power d+1 in B_j relates slice size to tangential moment. For any finite A\subset\mathbb Z_{\ge0}^d, let b=(d!\#A)^{1/d}. Among measurable subsets of the positive orthant of volume \#A, the simplex

minimizes the integral of u_1+\cdots+u_d. This follows directly by comparing the integrand inside and outside this sublevel set. Its integral is d b^{d+1}/((d+1)d!). On the other hand, the integral over A+[0,1]^d is \sum_{\beta\in A}(|\beta|_1+d/2). Enlarging d/2 to d, applying the resulting inequality slice by slice, and inserting the weights gives

5.4 The limiting contradiction

For each fixed center dimension d>0, the preceding bounds hold for every minimizing pair in the indicated ranges, with slice-growth constants independent of its center and degree. We now complete the proof of Proposition 5.1. By Propositions 3.2 and 4.4, there is a common interval of values of t immediately below n+1 on which the minima, for sufficiently large degrees, are bounded away from zero and one, and their discrepancies lie in a fixed compact interval [a_*,a^*]\subset(0,\infty). In particular,

Choose once and for all t<n+1 in that interval sufficiently close to n+1 that

Keep this choice of t in f_j=f_{j,t}(x). Put c=\liminf_{j\to\infty}f_j>0, and choose an increasing sequence of degrees m for which f_m\to c.

Suppose, seeking a contradiction, that infinitely many of these degrees admit a minimizing monomial pair with positive-dimensional center. Choose one such pair in each degree and pass to a subsequence on which the center dimension is a fixed d>0. The centers, valuations, and models are allowed to vary along this subsequence.

Apply the preceding construction to each chosen pair. Its test and coordinates may change with m, while all auxiliary degrees j for that m use that one test and coordinate system.

Fix an integer l\ge2. For 1\le i\le l, put j_i=\lfloor mi/l\rfloor, so j_l=m. Table 1 summarizes the dependencies and order of choices established above.

As m tends to infinity on our chosen subsequence, all increments j_{i+1}-j_i tend to infinity. Thus (5.15) applies successively. Starting with B_{j_1}\ge0 and then using (5.16) yields

The uniformity of C'_d is indispensable here, since the center and the model may depend on m.

For the auxiliary degrees, the test (j/m)v is admissible at x, because its center is still W. Its discrepancy is j\A(v)/m, and its section orders divided by j are v(s)/m. Applied to the basis with initials \Gamma_j, the definition of f_j therefore gives

This comparison occurs at x and does not require the chosen point y to satisfy specialization in degree j. Only the degree-m variation used (5.4).

Combining (5.19) with (5.7), we have

For each fixed i\in\{1,\ldots,l-1\}, the Hilbert asymptotic gives

Also j_i\to\infty, f_m\to c, and c=\liminf_j f_j>0, so

This does not require the degrees j_i themselves to belong to the subsequence realizing the liminf. Finally, \A(v)\ge a_* and t<n+1 imply

All summands in (5.18) are nonnegative for large m, and their number is fixed. Taking the liminf therefore gives

Only now let l tend to infinity. The right side tends to dI_*, which is strictly greater than d/t by (5.17). This contradicts (5.10).

There can consequently be only finitely many degrees on the selected subsequence admitting a positive-dimensional minimizing monomial center. A nonzero test has a nonempty closed center, and every center under consideration contains x. In all remaining degrees that center must therefore be exactly \{x\}, proving Proposition 5.1.

6 Isolation and adjoint lifting

We now use a point-centered minimizing pair to construct an effective rational SNC divisor whose coefficient equals the log discrepancy on a nonempty reduced divisor over x, and is strictly smaller on every other component meeting that divisor. This local configuration will allow Kawamata–Viehweg vanishing to lift a nonzero value in the fiber of P=K_X+(n+1)L at x.

6.1 Rational supporting data

Suppose henceforth, for contradiction, that P is not generated at x. Propositions 4.4 and 5.1 give fixed t<n+1 and m for which the infimum f_{m,t}(x) is attained and every minimizing monomial pair has center x. Choose such a pair (s_1,\ldots,s_{N_m};v) on a log resolution \pi:Y\to X of the divisors D_i=\operatorname{div}(s_i). Include all exceptional divisors in the SNC divisor on Y, and denote its components by E_j. Put

These vectors are rational and nonnegative, and a_j is a positive integer because X is smooth.

For every component Z of an SNC stratum whose closure has image containing x, let J(Z) be its set of crossing components and form the polytope

Only nonempty sets J(Z) are used. There are finitely many such polytopes. A convex representation z=\sum_j\lambda_jz_j is realized by the monomial weights \lambda_j/a_j; their discrepancy is one and their section-order vector is z. They can be rescaled to any positive discrepancy. If some weights vanish, the center enlarges to the appropriate stratum closure containing Z. Its image still contains x.

Write

The point q belongs to the union of the polytopes (6.1). It belongs to none indexed by a stratum with positive-dimensional image. Indeed, a representation in such a polytope, rescaled to discrepancy a, would give the same objective value and a center containing that positive-dimensional image. This would contradict Proposition 5.1.

Varying the scale of v differentiates the objective at its minimum. With

this derivative gives

For any C_Z containing q, vary the normalized profile along the segment from q to q'\in C_Z, holding the discrepancy equal to a. The one-sided derivative at the minimum is nonnegative. Since the objective decreases with each section order, this says

The coefficients needed for vanishing must be rational. We must approximate both q and b while preserving these supporting inequalities and the exclusion of every polytope with positive-dimensional image. The following elementary lemma preserves exactly which indexed polytopes contain the profile; the family may include repeated polytopes.

Apply Lemma 6.1, taking the neighborhood of b small enough to preserve the strict budget in (6.3). We obtain rational nonnegative q^0 and rational positive b^0 with

The profile q^0 belongs to no polytope with positive-dimensional image. In particular, q^0\ne0.

A rational point in a rational convex hull has a rational convex representation. The corresponding rational monomial weights, after clearing denominators, determine a primitive rational ray in the SNC crossing. Toroidal extraction of this ray gives a prime divisor F_0 whose normalized section-order vector is q^0. The monomial discrepancy formula gives equality in the retraction inequality of Lemma 2.2. Its center is x, by the exclusion of all positive-dimensional-image polytopes. We may retain this divisor on every subsequent common resolution: its orders, discrepancy, and equality under retraction to Y depend on the divisorial valuation, not on the higher model exhibiting it.

6.2 An isolated discrepancy-equality divisor

Let I=\{i:q_i^0>0\}, and choose an integer M>0 such that M/q_i^0 is an integer for every i\in I. Define the effective ideal

For any prime F, the order of this ideal divided by M is the minimum of the ratios \ord_F(D_i)/q_i^0, for i\in I. At F_0 all these ratios equal \A(F_0). A small contribution from this minimum will preserve equality at F_0. In the coefficient comparison below, equality in the perturbed bound will force these ratios to agree; positivity of every b_i^0 will also control the coordinates outside I.

Choose an integer u large enough that \mathfrak c(uL) is globally generated. Take a smooth projective resolution \rho:T\to X dominating Y, retaining F_0, and principalizing \mathfrak c:

Arrange that G, the pullbacks of the D_i, and all exceptional divisors have SNC support. The divisor u\rho^*L-G is globally generated, since it is the quotient of the pulled-back generating system of \mathfrak c(uL).

For a sufficiently small rational number 0<\eta<1, put

It is an effective rational divisor with SNC support. Since D_i\sim mL, we have

The first coefficient is positive when \eta is sufficiently small, by (6.5). Thus this difference is nef and big.

For a prime divisor F on T, write

Here orders of divisors on Y mean orders of their pullbacks. The number r_F is the discrepancy of the SNC retraction to Y. Let S be the reduced sum of the primes in the chosen SNC support satisfying all three conditions

The retained divisor F_0 is one of them, so S\ne0.

6.3 Lifting and descent

The remaining argument follows the discrepancy-and-vanishing strategy in [Fujita(1993), §1(1.6)], where strict discrepancy inequalities along the divisors meeting an equality divisor allow a nonzero fiber value to lift and descend. The construction above supplies that local configuration on the reduced divisor S.

Define the integral Cartier divisor

Its difference from the canonical divisor with fractional boundary is

The fractional part \{B\} is an effective rational SNC boundary with all coefficients in [0,1), and the right side is nef and big by (6.8). Theorem 2.4 therefore gives H^1(T,\cO_T(D))=0. The exact sequence of the reduced Cartier divisor S gives a surjection

Compare the divisor upstairs with the desired adjoint bundle:

On a component F of S, Lemma 6.2 and the integrality of a_F give

For every other prime meeting S, the same lemma gives \lfloor\operatorname{coeff}_F B\rfloor\leq a_F-1, and hence \operatorname{coeff}_F C\geq0. Primes outside the chosen SNC support contribute zero. Thus C is effective on a neighborhood of S, with no component of S in its support. Negative components of C may occur elsewhere, but their supports are disjoint from S; removing those finitely many supports gives the asserted neighborhood.

Since S is reduced and each of its components maps to x, the restriction \rho|_S factors scheme-theoretically through the point x. Indeed, the pullback of a function vanishing at x vanishes on every component of S, and therefore vanishes on the reduced scheme. It follows that

Choose a nonzero element of the one-dimensional fiber P|_x. Multiply the corresponding constant section on S by the canonical rational section of \cO_T(C). This latter section is regular near S, because C is effective there, and is nonzero at the generic point of each component of S. The product is therefore a global section of \cO_S(D+S), nonzero at those generic points. Notice that no global effectiveness of C is required for this restriction construction.

Lift this section using Equation (6.14). The positive part of C is exceptional over X: on a nonexceptional prime not belonging to S, the coefficient of C is -\lfloor\operatorname{coeff}_F B\rfloor \leq0, while on a component of S it is zero. Since S is nonempty and every component maps to x, any such component, together with the imposed generic nonvanishing, satisfies the hypotheses of Lemma 2.5. The lift therefore descends to a section s\in H^0(X,P) with s(x)\ne0.

The argument also includes dimension one. In that case a component of S can be the nonexceptional point x itself; its coefficient in C is still zero, and the same restriction and descent argument applies.

We have contradicted the assumed failure of generation at x. Since a failure of global generation would occur at a closed point, this proves Theorem 1.1.

6.4 All adjoint powers