Fujita’s freeness conjecture

1. Lifting in the equality case makes the mean order strictly larger

Let $H_j$ be the initial-exponent set for the Taylor monomial order comparing ordinary degree first. From $\#H_j=N_j$ and the Hilbert asymptotic, filling lattice points in increasing total degree gives

If the bound with right-hand side $n/(n+1)$ were an equality under failure of generation, then $L^n=1$, and the normalized initial exponents would eventually meet every fixed small ball in the interior of the simplex. Choosing $n<t'<n+1$, the manuscript forms products of sections whose leading forms define an ideal with threshold greater than $1/h$ on the exceptional divisor $E$ of the point blowup. Degeneration to monomials and semicontinuity from Singularity exponents [DK] transfer this strict inequality back to the original leading forms.

After resolving the ideal, the remaining positivity is $(n+1-t')L$. Theorem 2.4 kills $H^1$, a nonzero fiber value on $E$ lifts, and Lemma 2.5 descends it to a section nonzero at $x$. This contradicts failure of generation. Thus Proposition 3.2 gives the strict inequality $\liminf>n/(n+1)$.

Proposition 3.2 · pp. 6–9

Singularity exponents [DK] · Theorem 3.1; Lemma 3.2 · pp. 14–15

2. Fix a basis through a flag before taking a minimum on an SNC model

For an ordered degree-$m$ basis $\mathbf s=(s_i)$ and $t>0$, set

Proposition 3.2 makes the derivative for $v=\lambda\operatorname{ord}_x$ negative at $\lambda=0$. Lemma 4.2 gives $0<c\leq f_{m,t}(x)\leq1-\delta$ for a common interval immediately below $n+1$ and all sufficiently large $m$. On the relevant sublevel set it also gives $0<a_*\leq A(v)\leq a^*<\infty$. The upper bound uses a supply of low-order jets forcing a fixed proportion of every basis to have low order.

(4.1) · p. 9

The proof does not compactify all bases or all models at once. Rational lower bounds on normalized section orders become a threshold condition for a sum of powers of base ideals. Lemma 2.3 gives closed loci in the complete flag variety. Choose a flag in the nested intersection and a basis adapted to it; only then resolve this basis’s divisors on one model. SNC retraction preserves section orders and does not increase discrepancy, while normalized weights lie in a compact simplex. Weights may vanish in the limit, but the center then enlarges and still contains $x$. This yields the minimizing pair of Proposition 4.4.

Lemmas 4.2–4.3; Proposition 4.4 · pp. 10–12

SNC resolution [RES] · Theorem 1.0.1 · print p. 781 / PDF p. 3; Definition 2.1.3 · print p. 783 / PDF p. 5; Theorem 2.4.1 · print p. 786 / PDF p. 8

3. Force every minimizing center to be a point and rationalize support data

Proposition 5.1 fixes one $t<n+1$ for which, in arbitrarily large degrees, every minimizing pair has center $\{x\}$. This universal assertion is needed next: another stratum realizing the same normalized section-order profile and discrepancy would give the same minimum.

On an SNC model of $D_i=\operatorname{div}(s_i)$, normalized profiles from the strata form finitely many rational polytopes. The gradient of the objective gives positive supporting coefficients $b_i$, and differentiation in the scaling direction gives $m\sum_i b_i=t<n+1$. Lemma 6.1 rationalizes the profile and supporting coefficients without changing membership in the polytopes or losing this strict budget. The ideal perturbation of Lemma 6.2 then isolates a nonempty reduced discrepancy-equality divisor $S$ over $x$.

Proposition 5.1 · p. 13

§6.1; Lemma 6.1; (6.1)–(6.5) · pp. 19–20

Lemma 6.2 · pp. 21–22

4. A local coefficient comparison gives a section nonzero at the original point

On a resolution $\rho:T\to X$, the effective rational SNC divisor $B$ leaves $(n+1)\rho^*L-B$ nef and big. Put $D=K_T+(n+1)\rho^*L-\lfloor B\rfloor$; then $D-(K_T+\{B\})$ is this remainder. Kawamata–Viehweg vanishing [KV] gives $H^1(T,\mathcal O_T(D))=0$, hence a surjection

The comparison divisor is

By Lemma 6.2, its coefficients are zero on $S$ and nonnegative on other components meeting $S$. Thus $C$ is effective near $S$ and has no component of $S$ in its support. Since $S$ is reduced and all its components map to $x$, one has $\rho^*P|_S=P|_x\otimes\mathcal O_S$. Multiplying a nonzero fiber value by the canonical rational section of $C$ gives a section nonzero at the generic point of every component of $S$. It lifts by the displayed surjection.

On nonexceptional components the coefficients of $C$ are nonpositive, so its positive part is exceptional. Lemma 2.5 descends the lift to $X$ with $s(x)\neq0$. For $n=1$, a component of $S$ may be the point $x$ itself, and the same coefficient comparison applies. The assumed failure at any point has been contradicted, completing the global-generation conclusion of Theorem 1.1. Only the comparison near $S$ is needed: neither global log canonicity of $B$ nor global effectivity of $C$ is required.

§6.3; (6.13)–(6.15) · pp. 22–23

Theorem 2.4 · p. 5 · Lemma 2.5 · p. 6

Kawamata–Viehweg vanishing [KV] · Theorem 0.1 · p. 1 · Adjoint lifting [FL] · §1(1.6) · p. 3

1. Separate the restriction-image bound from the center’s degree and singularities

Fix a very ample multiple $kL$. For any integral $d$-dimensional subvariety $W$, put $I_W(h)=\dim\operatorname{im}(R_h\to H^0(W,hL|_W))$. Lemma 5.2 gives

with $C_d,H_d$ independent of $W$. Surjectivity of restriction is not used. A general linear projection $W\to\mathbf P^d$ has degree $D=k^d(L^d\cdot W)\geq D_0=k^d$. Bounded-degree ambient forms separate $D_0$ points of its geometric generic fiber and yield $D_0$ elements independent over $\mathbf C(\mathbf P^d)$. Multiplication by monomials from the base gives $I_W(qk)\geq D_0\binom{q-D_0+d}{d}$. Globally generated multiples handle the finitely many residue classes and cover all degrees. The argument uses the fixed number $D_0$ of points, not an upper bound on $D$.

Lemma 5.2; (5.1)–(5.3) · pp. 13–14

2. Use specialization to make the tangential derivative nonnegative

Suppose a minimizing pair has center $W$ of dimension $d>0$. Lemma 4.5 gives $f_{m,t}(y)\geq f_{m,t}(x)$ at a very general smooth point $y\in W$. On the stratum above it, split coordinates into the old positive-weight directions, fiber directions and tangential directions from $W$. Write initial exponents as $(p,\beta)$ and the old weight as $\ell(p)$. Replacing the basis by one adapted to this initial order preserves minimality.

Give weight $\epsilon$ to the $d$ tangential coordinates and $\epsilon^2$ to the remaining fiber coordinates. For each fixed basis, $A(v_\epsilon)=A(v)+d\epsilon+O(\epsilon^2)$ and $v_\epsilon(s)=\ell(p)+\epsilon|\beta|_1+O(\epsilon^2)$. The perturbed center is $y$, so specialization is what makes $F(v_\epsilon,\mathbf s)\geq f_{m,t}(y)\geq F(v,\mathbf s)$ available. Consequently

The denominator in the exponential defining $Q_j$ is the reference degree $m$, not the auxiliary degree $j$. Perturbation and differentiation are performed at fixed $m$; no choice of $\epsilon$ uniform in all degrees is required.

Lemma 4.5 · pp. 12–13 · §5.2; (5.4)–(5.10) · pp. 14–16

3. Weight the growth of sumsets to contradict the upper bound

For a fixed transverse exponent $p$, consider the slice $G_p(j)$. Restricted sections supply enough points in $G_0(h)$, and in the fixed degree $h_0=kd$ one also has $\{0,1\}^d\subset G_0(h_0)$. Thicken finite sets by unit cubes and apply Brunn–Minkowski [BM]. For nonempty slices, $b_p(j)=(d!\#G_p(j))^{1/d}$ satisfies

where $C'_d$ is independent of the center, test and $j,p$. Weight this inequality by $e^{-\ell(p)/m}$. Convexity of the $(d+1)$st power and a simplex first-moment estimate turn it into a lower bound for $T_m$.

§5.3; (5.11)–(5.16) · pp. 16–17 · Brunn–Minkowski [BM] · Theorem 4.1 · print p. 362 / PDF p. 8

The uniform $a_*>0$ from Lemma 4.2 gives

Fix $t<n+1$ once so that $I_*>1/t$. Choose a subsequence along which $f_{m,t}(x)$ approaches its positive liminf. If positive-dimensional centers occur infinitely often, pass further to a fixed dimension $d$. At auxiliary degrees $j_i=\lfloor mi/l\rfloor$, the rescaled test $(j_i/m)v$ is admissible at $x$, so specialization at $y$ is not needed in every auxiliary degree. First let $m\to\infty$ for fixed $l$, and only then let $l\to\infty$. The result is

contradicting the first-variation bound. Far enough along this subsequence, no minimizing pair has a positive-dimensional center. This proves the universal assertion in Proposition 5.1, stronger than existence of one point-centered minimizing pair.

§5.4; (5.17)–(5.21) · pp. 17–19

1. Rationalize support conditions and point-center constraints together

On each polytope containing the minimizing profile $q=(v(s_i)/A(v))_i$, the gradient gives $b\cdot z\leq b\cdot q=1$. If $q$ also belonged to a positive-dimensional-image polytope, it would realize a minimizing pair with a nonpoint center, contradicting Proposition 5.1. Lemma 6.1 chooses a rational point in the intersection of the smallest faces containing $q$, while avoiding the finitely many polytopes not containing it. After obtaining $q^0$, it chooses $b^0>0$ in a rational feasible polyhedron while preserving $m\sum b_i^0<n+1$.

Toroidal extraction of the rational monomial ray gives $F_0$. Its profile is $q^0$, its retraction discrepancy inequality is an equality, and its center is $x$. This divisor is retained in subsequent resolutions.

§6.1; Lemma 6.1; (6.1)–(6.5) · pp. 19–20 · §2.1; Lemmas 2.1–2.2 · pp. 3–4

2. Mixing in a minimum preserves the positivity needed for vanishing

The order of a sum of ideals is the minimum of their orders, hence

Meanwhile $u\rho^*L-G$ is globally generated. Since $D_i\sim mL$,

For small $\eta$ the first coefficient is positive, the second summand is nef, and the total is nef and big. A component of $S$ has retraction profile $q^0$. Its retraction center has image containing $x$; exclusion of positive-dimensional-image polytopes forces every such component to map to $x$. Both the minimum above and the pairing with $b^0$ equal $a_F$, giving $\operatorname{coeff}_F B=a_F$.

§6.2; (6.6)–(6.9) · p. 21 · (6.10)–(6.12) · p. 22

3. Equality on an adjacent component forces membership in S

If $F$ meets a component $F_S$ of $S$, both retraction profiles lie in the same $C_Z$ determined at an intersection point. For $r_F=0$, all section orders and the order of $G$ vanish, so $\operatorname{coeff}_F B=0<a_F$. For $r_F>0$, put $q'=(\operatorname{ord}_F(D_i)/r_F)_i$ and $\mu=\min_{i\in I}q_i'/q_i^0$. Then

Equality forces $r_F=a_F$ and $b^0\cdot q'=\mu=1$. The positively weighted average on $I$ equals its minimum, so $q_i'=q_i^0$ there. Strict positivity of $b_i^0$ outside $I$ also forces $q_i'=0$ there. Thus $q'=q^0$. Since the intersection point lies over $x$, the prime $F$ satisfies every defining condition for $S$. All adjacent components outside $S$ therefore have strict inequality. This local comparison feeds into the main theorem’s lifting step.

(6.10)–(6.12) · p. 22

1. Adjust the dimension and adjoint bundle together

Set $r=m-n-1\geq0$, $Y=X\times\mathbf P^r$, and $A=L\boxtimes\mathcal O_{\mathbf P^r}(1)$. Then $Y$ is smooth, connected and projective over $\mathbf C$, with $\dim Y=n+r$, and $A$ is ample. Theorem 1.1 applied to $(Y,A)$ makes

globally generated.

Corollary 6.3 · p. 23

2. Restrict to a fiber to recover every m

The restriction of a globally generated line bundle is globally generated. Restricting to $X\times\{z\}$ therefore gives global generation of $K_X+mL$. The case $r=0$ is Theorem 1.1 itself. Since $m\geq n+1$ was arbitrary, this proves the entire conclusion of Corollary 6.3.

Corollary 6.3 · p. 23