Quadratic generation
for smooth polytopes with long edges

Caius Folkerts

Abstract. Every smooth lattice polytope of dimension

whose edges have lattice length at least

is normal and has a toric ideal generated by quadrics. Thus a single edge bound gives both projective normality and quadratic equations in every dimension. We prove two mixed decomposition theorems; an interior point of a sum admits every prescribed parity in its first summand at edge bound

, and multiplication by an adjoint factor is surjective at edge bound

. The common mechanism is a lattice-covering theorem for averages of basis boxes. Large projective-space fibers account for the failure of the requisite boxes; their disjointness permits a simultaneous correction. A general exchange argument then reduces quadratic generation in all degrees to the creation of a repeated summand in degree three.

1. Introduction

A smooth lattice polytope determines a projective embedding of a smooth toric variety, but smoothness alone does not yet provide a general description of the generators and relations of its homogeneous coordinate ring. The normality question associated with Oda and the quadratic-generation question associated with Bøgvad ask whether these generators always have degree one and these relations always have degree two; see, for example, [2]. We establish both conclusions under an edge bound depending only on the dimension.

Let , let be a lattice polytope, and write . The polytope is smooth if the primitive directions of its edges at each vertex form a basis of . It has the integer decomposition property if every point of is a sum of points of , for every integer . Over an arbitrary field $𝕜$ , its toric ideal is

𝕜 𝕜

Theorem 1.1. Let be a smooth -dimensional lattice polytope, where . If every edge has lattice length at least , then has the integer decomposition property and

The assertion is independent of the ground field. In the language of projective embeddings it is property ; the embedding is projectively normal and its homogeneous ideal is generated by quadrics. Bounds on powers of line bundles and bounds on the individual edges are different kinds of positivity; the theorem uses only the latter. Recent work of Song and Wen [3] gives the bound for property under additional conditions on the fan. Theorem 1.1 gives the case on every smooth projective fan.

The proof requires more than the existence of integral decompositions. To change one decomposition into another, we must control the residue class of a summand, including when the target approaches a face. The following mixed statements provide that control. If is an ample integral divisor on a smooth projective toric variety, write for its lattice polytope; the degree of on an invariant curve is the lattice length of the corresponding edge.

Theorem 1.2 (Prescribed parity). Let be ample integral divisors on a smooth projective toric -fold, each having degree at least on every invariant curve. For every and , there are and such that and .

Theorem 1.3 (Adjoint multiplication). Let be ample integral divisors on a smooth projective toric -fold, each having degree at least on every invariant curve. Then

Equivalently, multiplication of sections of and onto those of is surjective.

The distinction between the two bounds already appears for a simplex. The polytope contains the unit boxes at its vertices, but an interior lattice point first appears in . In the proof, boxes of side two enforce parity, while a unit box inside an adjoint polytope supplies an interior summand. A simplex factor that is too short to contain the desired box is treated by distributing its homogeneous coordinates directly.

Section 2 explains why these are the right decomposition statements by deriving quadratic generation from an exchange criterion for arbitrary lattice polytopes. Sections 3 and 4 develop the toric geometry and lattice-covering lemma. A corner box can fail along a contractible projective-space fiber of dimension greater than , but distinct such loci are disjoint. Section 5 uses this fact to construct corrected boxes, and Sections 6 and 7 prove the mixed theorems by induction on dimension.

2. From parity to quadratic generation

A degree- decomposition of is a multiset of points of with sum . A quadratic exchange replaces two entries by two other points of having the same sum. The following criterion isolates the algebraic part of the argument.

Proposition 2.1. Suppose has the integer decomposition property and the following properties, in the saturated affine lattice of each face

(i): for every face , every lattice point of admits a decomposition in with any prescribed parity of the first summand;
(ii): for every proper face , every lattice point of is the sum of a lattice point of and a lattice point of .

Then every decomposition fiber is connected by quadratic exchanges, and is generated by quadrics over every field.

Proof. We first change any triple into one with a repeated entry. Let be a triple and its smallest containing face. If some pair, say , already has smallest containing face , then . By (i), write with modulo twice the face lattice. The midpoint is integral and belongs to , so

are quadratic exchanges.

Otherwise choose a pair with smallest containing face . If the pair repeats, there is nothing to prove; if it does not, (ii) changes it into with . Every face containing contains , so and the third entry have smallest containing face . The preceding argument applies. Here is a proper face of , even when , which is precisely where condition (ii) is needed.

To connect two repeated triples with common sum, write

Then , so the two exchanges

connect the triples. Thus degree-three fibers are connected. For , induct on and create a repeated entry in each of two decompositions

The point

has an integral decomposition . Holding one fixed, induction connects the first state to ; holding fixed connects this state to the second. Finally, distinct target characters are linearly independent, so the kernel is spanned by differences of monomials in common fibers. Along an exchange path each such difference telescopes into monomial multiples of quadratic binomials. □

Theorem 1.2 supplies (i) for every face of the polytope in Theorem 1.1. A proper face of dimension inherits the edge bound , so Theorem 1.3, applied to that face twice, supplies (ii). It remains to prove the mixed statements and the integer decomposition property.

3. Contractible curves and local boxes

Throughout the geometric proof, is a smooth projective toric -fold with character lattice and cocharacter lattice . Write for its primitive ray generators and for the invariant divisors. For a real invariant divisor , set

The Cartier point at a maximal cone has equality on its rays. Nefness means that all these points belong to . For nef real divisors,

These formulas include lower-dimensional nef polytopes. Integral nef divisors have integral Cartier points, and .

3.1. The contraction data. We use Casagrande’s description of contractible classes and their behavior under smooth invariant blowups [1]. A normalized extremal class is the class of an actual invariant curve. It is contractible, and every contractible class is represented by an invariant curve and has a primitive relation

For the right side is zero. The positive rays form a primitive collection; every proper subset spans a cone, whereas the full set does not. Intersections with invariant divisors are the displayed coefficients, so

An edge bound therefore gives for every contractible class. Nefness, on the other hand, is tested on normalized extremal classes.

If , the contraction is a bundle over a smooth projective toric base of dimension . If , its exceptional locus is

a bundle. The normal line bundles supplied by the restrict to on each fiber. The base is projective, because the fiber class on the projective variety is numerically effective [1, Corollary 3.1].

For a smooth invariant blowup with exceptional divisor , every contractible class other than a blowup-fiber line pushes to a primitive relation, and pushforward is injective on these classes. If the pushed class is not contractible, the original class has -degree one [1, Theorem 5.1]. For center (3), the unique contractible lift of its fiber class has relation

with zero coefficients omitted. Its -degree is , as follows from the type-a case of [1, Lemma 5.4]. For disjoint centers these statements apply successively, with intersections against the other exceptional divisors unchanged.

3.2. Coordinates on the projective bundles. The bundles above split equivariantly. Any fiber rays extend to a smooth maximal cone, hence span a saturated sublattice. Smoothness of the bundle gives one primitive lift of each base ray, and the contraction fan contains every proper fiber cone over every base cone. After an integral splitting of the lattice sequence, the fiber coordinates of the lifted rays are integral piecewise-linear functions on the base fan. They define line bundles , one normalized to zero, with bundle

Use the convention that an integral homogeneous fiber weight , , of a divisor of fiber degree has base divisor

The endpoint divisors restrict to the coordinate sections, so their polytopes are faces of . For ,

Thus edge bounds and primitive-relation inequalities verified at the endpoints hold at every actual integral weight. Adjunction takes the useful form

In a multiplication problem, the sum of the base-weight divisors depends only on the total fiber weight.

The normal-height, fiber, and base coordinates are integral and saturated. The normal rays of a smooth face form part of a lattice basis, so the height map onto is surjective and its kernel is the face lattice. The bundle lattices form a split exact sequence. Prescribing parity in these successive coordinates therefore prescribes parity in .

3.3. Two local criteria. At let be the primitive outgoing edge basis. Its corner box of side is

Lemma 3.1 (Corner criterion). An ample real divisor contains all its corner boxes of side if for every extremal relation (2). Conversely, containment implies this inequality for every contractible class.

Proof. For put . A cone contains at most positive rays of a primitive collection, and negative coefficients can only decrease . Hence is nef. Its Cartier point on is the box corner and belongs to .

Conversely, the contraction fan has a maximal cone containing all negative rays and all but one positive ray of a contractible relation. At the associated vertex the slack of the missing-positive-ray inequality is ; at the box corner selecting the other positive rays it is . Containment makes this nonnegative. □

Faces are smooth in their saturated lattices and inherit contained corner boxes. In particular, edge bound gives unit boxes in dimension .

Lemma 3.2 (Displaced slices). Suppose every invariant degree of an ample real divisor is at least . If and , then is nef, is ample when , and has every invariant degree at least . Its restriction to in (3) is the exact slice of at normal heights , with fiber degree .

Moreover, if a smooth real polytope has all edges of length at least , every vertex has height at least above each nonincident primitive facet.

Proof. A positive prime-divisor intersection with an extremal class is one, so is nef. The expression

proves all positivity and degree assertions, including for nonextremal invariant curves. The face defined by the normal equalities in is the restriction polytope, proving exactness; the fiber degree follows from (2).

At a vertex outside a facet, an outgoing edge decreases the facet height, since otherwise the tangent cone would make that vertex a global minimum. Its primitive height slope is a negative integer and its length is at least , giving the last assertion. □

4. Averages of lattice basis boxes

The geometric arguments will place a target in a convex hull of matched box centers. The next lemma converts this real-valued information into a lattice decomposition; the bases of the different boxes may be unrelated.

Lemma 4.1 (Lattice covering). Let be vectors of a full lattice spanning . If is a convex combination of incidence vectors of subsets that are real bases, then

meets every translate of . In particular, every convex Minkowski average of centered integral basis boxes is lattice covering.

Proof. We choose an ellipsoidal norm so that comparison with neighboring lattice points places every nearest-point error inside . The required balance is supplied by a determinant minimization.

Let be the bases of the configuration and . First take , put , and minimize on the function

Cauchy–Binet expands the determinant as . Relative interior gives with

Thus is coercive and attains its minimum. The logarithmic gradient lies in , so vanishing derivatives on give

For any , choose a nearest lattice point in the -norm. Comparison with yields

Since ,

For boundary , approximate by relative-interior points. The zonotopes converge and remain bounded, so the corresponding lattice points have a constant subsequence; passage to the limit finishes the proof. □

Suppose contain boxes with the same basis at each maximal cone and centers . If , the common centered average satisfies

Unit boxes and give a lattice decomposition. Side-two boxes and give every parity class. This explains why the centers, as well as the boxes, must fit into a common nef-divisor construction.

Lemma 4.2. If two ample integral polytopes on the same smooth fan contain their unit corner boxes, then

Proof. The center at is . For a primitive facet normal , its slack is at least , because the box is contained and the final sum is a positive integer. Thus these centers are feasible Cartier points for the nef real divisor ; likewise for .

An interior lattice target has every facet slack at least one and therefore lies in

Equation (1) expresses it as an average of matched centers, so (8) and Lemma 4.1 apply. A boundary target reduces to matched proper faces, which inherit the smooth fan and unit boxes; induction on dimension completes the proof. □

Corollary 4.3. Every smooth -polytope with edges of length at least has the integer decomposition property.

Proof. The corner criterion gives unit boxes. Apply Lemma 4.2 to and , and induct on . □

The next allocation replaces a missing box on a simplex fiber.

Lemma 4.4 (Simplex parity). For and , every interior lattice target in has a decomposition in each prescribed parity of its first factor.

Proof. Write the homogeneous target as , . Let be the requested and complementary parities, including the zeroth coordinates determined by the degrees. Then , and the parity-vector sums agree with modulo two. Also ; when , parity excludes ones, and when the bound is automatic. The same holds for . The intervals

therefore admit total ; subtract their lower bounds and divide by two, then distribute the resulting integral total among the integral capacities. □

5. Large contraction loci and corrected boxes

The corner criterion shows that a side-two box can fail only along a contractible fiber of dimension greater than . Two such fibers cannot meet if their numerical classes differ; an intersection would have positive dimension and contain a curve whose class is a positive multiple of both classes. Thus the loci

are pairwise disjoint. The same argument shows that a class with is the only class in any collection of this kind.

Blow up any subcollection with . For make no modification and write . Over a point of , the divisor is

With the convention of lines, is the negative tautological line bundle; its dual is ample on this fiber. Thus has negative degree on every curve contracted by , and the remain disjoint.

Lemma 5.1. Let be a normalized extremal class on the resulting variety, with positive rays and negative rays.

(i): If , then .
(ii): If both have positive degree, their two rays form the entire positive collection, and , .
(iii): Every maximal cone containing contains exactly rays of the original positive collection .

Proof. For (i), a contraction fiber meets in a coordinate hyperplane on which has degree one. Its map to has no positive-dimensional fiber, since a curve in one would have both positive and negative -degree. Hence .

For (ii), disjoint divisor rays are a noncone pair, so minimality makes this pair the entire positive collection. The contraction locus is a bundle over of dimension ; for use the whole variety. The two divisors give disjoint sections with dual normal line bundles. The joint map has no positive-dimensional fiber, since both dual bundles would be negative on a curve in it. Thus .

For (iii), suppose an old maximal cone containing the negative rays contains rays of . Primitivity gives . Remove these rays and the negative rays, leaving on rays. The contraction fan supplies a cone containing , all negative rays, and any positive rays; its rays force . Star subdivision preserves these positive rays. □

5.1. Original boxes away from the loci. We must select all obstructing contractible classes here, because an extremal class after blowup may push to a nonextremal contractible class.

Proposition 5.2. Let all invariant degrees of an ample divisor be at least . Select every contractible class with , and suppose none has . At every original vertex outside their loci, the corner box of side two is contained in .

Proof. Fix its maximal cone and . The desired box corner is the Cartier point of , where . We will make this divisor nef by subtracting exceptional divisors over the selected loci, leaving its Cartier point at unchanged. The correction must repair each negative fiber degree without creating a negative degree on a different curve.

Activate the selected indices with , and write

Since is outside ,

Blow up the activated loci of codimension at least two, retaining divisors in codimension one. We claim that

is nef. A blowup-fiber line has degree . Consider any other normalized extremal class .

If no is positive, successive pushdown gives an old contractible class . For an inactive class, ; use for an unselected class and inactivity otherwise. All exceptional corrections help. For an activated class, injectivity identifies the unique lift (4); its degree is , with zero intersections against other .

If exactly one is positive, the image of an invariant representative is a nonzero effective multiple of an old invariant curve, so . Since , Lemma 5.1 gives

If two are positive, then and neither positive ray is in . Put , , , and . Let be the sum of negative wall-coefficient magnitudes on old rays in . Discarding other helpful terms gives

The bounds (9) first give . For a complementary estimate, each has all but one ray in an invariant wall representing , so their intersection has at most one ray outside the wall. By Lemma 5.1(ii), at most wall rays have zero coefficient. Consequently

Together with , this gives . The two lower bounds sum to two, so one is positive. Three positive exceptional rays are impossible, because any two already form a primitive noncone pair. This proves nefness.

The unchanged cone contains no exceptional ray, so its Cartier point for is . It belongs to , since (10) subtracts effective divisors. Varying supplies every box corner. □

6. Prescribing the parity of a summand

We prove Theorem 1.2 by induction on . Dimension zero is immediate. For intervals, an interior target gives a feasible first-summand interval with integral endpoints and length at least one, so both parities occur. For , Lemma 4.4 applies.

Select the union of all contractible classes on which at least one of has degree less than . Every selected exceeds , so the loci are disjoint. We first treat a possible fiber-type class, then build boxes and their centers on a common blowup. Targets outside the sum of the center polytopes will lie close to an old facet or a selected locus, where induction applies.

6.1. A fiber-type class. If a selected class has , it is the sole selected class and is a split bundle over a base of dimension . Both fiber degrees are at least . Lemma 4.4 splits the target’s homogeneous fiber coordinates in the prescribed parity. Each resulting base-weight divisor has edge degrees at least , inherited from the coordinate-section faces through (6). Their sum depends only on the total weight, and the base target is interior because the original target satisfies the corresponding facet inequalities strictly. Induction in dimension gives the base parity, which combines with the fiber parity to give the prescribed class in . Henceforth every selected . Blow up their disjoint loci as in Section 5.

6.2. Shifted boxes and their centers. For , set and

Both factors use the same selected loci, even if one had no obstruction there. Since and ,

Define the real center divisor

We check nefness first, since it will allow all the boxes to be averaged. A blowup-fiber line has degree . Every other invariant representative has -degree at least . If no selected divisor is positive, the degree is at least , where are negative-coefficient magnitudes; here , since . With one positive , Lemma 5.1 gives

With two positive divisors, and

Thus is nef.

Outside the selected loci, Proposition 5.2 supplies the original side-two boxes, centered at the Cartier points of . At a vertex of , let be the old primitive normal directions and complete them by tangent directions. On the new cone indexed by , the normal basis is

Take the side-two box based at in this basis, with the unchanged tangent directions. Its normal heights satisfy

the lower bounds use .

By Lemma 3.2, the tangent slice is the restriction of to . Its fiber degree is at least , while every other edge degree is at least . In a split bundle, every nonfiber primitive collection lifts a base primitive collection and has at most rays. The corner criterion therefore supplies all tangent side-two boxes, proving containment of the full shifted box. For , only the fiber inequality is needed, even at the nef endpoint .

Its base point is the Cartier point of , so its center is exactly the Cartier point of . When , this is simply a normal shift in the original basis.

6.3. Targets covered by the centers. For an interior lattice target , let be its primitive facet slacks. We have whenever

For , the second condition is the exceptional-ray inequality; for , it strengthens the selected old-facet inequality. Nefness expresses as an average of matched center sums. The contained side-two boxes and Lemma 4.1, with , give every required parity.

6.4. Targets near an old facet. If at an old facet , the desired normal parity chooses heights or . The two slices are ample integral polytopes on the smooth facet fan, with edge bounds at least and by Lemma 3.2. Their sum is the exact first slice of , and the target is interior in it. Induction in dimension supplies the tangential parity, which combines with the chosen normal parity by saturation of the face lattice.

6.5. Targets near a selected locus. The remaining case is a selected locus with total normal height

Put . The shift bounds give

We need a split in the prescribed normal parity with and . Let be the desired and complementary parities. Positivity of gives . Set

The capacities for and are

They are nonnegative because , and (18) implies . Allocate the integral tokens among the coordinate capacities so that , , and . This gives the required normal split.

The exact slices at are nef restrictions to , ample there when , and are split -bundle polytopes. Their fiber degrees are at least . The target is interior in their sum; the sum divisor is the restriction of , which is ample because , and its incident facet slacks are the original positive slacks. Split the homogeneous fiber coordinates in the desired parity by Lemma 4.4. Each resulting base weight has edge bound at least or , and their sum depends only on the fixed total coordinates. Induction in dimension supplies the base parity, with no further step when .

These cases exhaust the targets, proving Theorem 1.2.

7. An interior summand and adjoint multiplication

We prove Theorem 1.3 by a separate induction on dimension. The geometry follows the parity argument, but the second factor must remain inside , changing the displacement and the center divisor. Dimension zero is immediate. On , homogeneous-coordinate allocation applies to simplices of degrees and , including a point as the second simplex.

Assume . Then is ample; every extremal positive collection has at most rays, so its degree is at least .

7.1. Selected loci and inner slices. Select all contractible classes with

The bound gives , so their loci are disjoint.

We need the following exact inner-slice description. For , , and , the divisor is ample with edge bound . If every , the slice of at these original normal heights is

For incident facets this is adjunction, . A nonincident facet has slack at least at every vertex of the displaced face, by Lemma 3.2; hence the canonical inequalities make no further cut. This also proves exactness at the real heights used below.

If a selected class has , then is a split bundle over a base of dimension . Its two original fiber degrees are at least , and the adjoint second degree is . Allocate the integral fiber coordinates between an weight and an adjoint weight . By (7), the base target has divisor . The two parent base divisors have edge bound , inherited from coordinate-section faces, so induction in dimension supplies the base decomposition. We may now assume every selected .

7.2. Inner unit boxes away from the loci. Fix a maximal cone outside all selected loci and . Set

As , every class on which is negative is selected. Activate those with and set

Then

Blow up the activated centers and put . A blowup-fiber line has degree . With no positive exceptional divisor, the pushdown and unique-lift argument of Proposition 5.2 gives nonnegative degree.

For the other cases, use

With one positive , Lemma 5.1 yields

With two positive divisors, and

Thus is nef. Its Cartier point on the unchanged cone is

and belongs to because the correction subtracts effective divisors. Varying supplies the unit box based at the Cartier point of . The ordinary unit boxes are contained everywhere by the corner criterion.

7.3. Shifted unit boxes. On the common blowup of all selected loci, put

at a locus with parameters . Then

For the first factor, use the new basis (15) and the unit box based at the Cartier point of . Its heights satisfy and . In the exact tangent slice , the fiber degree is at least , and every other edge degree is at least . The corner criterion supplies every tangent unit box, so the full box lies in .

For the second factor, take the unit box based at the Cartier point of . On the cone indexed by , its initial normal heights are for and . Throughout the box,

The tangent slice is by (20). Before adjunction, its fiber degree is at least

so the inner fiber degree is at least . Every other parent edge degree is at least . For a nonfiber extremal collection of size , adjunction leaves degree at least . Thus the corner criterion supplies the tangent boxes, and the full second box lies in .

7.4. Nefness of the centers. The centers of the matched boxes are Cartier points of

At unchanged cones these are also the centers of the preceding boxes. We verify nefness in the same three cases of Lemma 5.1.

For , a blowup-fiber line has degree . Every noncontracted extremal representative has -degree at least . With no positive selected divisor, the bound suffices. With one positive ,

With two it is at least , because .

For , a blowup-fiber line has degree . If no selected divisor is positive, rewrite

The correction coefficients are nonnegative, and the class pushes to an old contractible class. For a nonselected class, failure of (19) gives

For a selected class, its unique lift and (22) give

With one positive ,

With two, the bounds on and yield

Both center divisors are therefore nef.

7.5. Decomposing the targets. Take and write for its primitive heights relative to . If

then . At an exceptional ray the two center heights are and , explaining the second threshold. Averaging matched centers and applying Lemma 4.1 to their unit boxes produces a lattice point in .

If at an original facet , the first factor has height zero and the inner second factor height one. The parent slices and have edge bounds and . Adjunction and the nonincident-facet reserve identify the inner second slice with , so dimension- induction decomposes the tangential target.

Finally, suppose all old heights are at least two but (26) fails at a selected locus. Then satisfies . Give the second parent heights and the first heights . Thus , , and , and both displaced parent slices are ample and exact. Put , . The remaining target belongs to . The second adjoint divisor is nef; its fiber degree is

and every other extremal collection has size , giving degree at least .

Allocate the target’s homogeneous fiber vector as , with equal to the fiber degree of . Integral allocation is possible by distributing this total among the capacities . The base target has divisor

Its two parent base weights are ample on the same fan, with edge bounds at least and . Induction in dimension supplies the base decomposition and completes the allocation in all coordinates. This proves Theorem 1.3.

Proof of Theorem 1.1. Corollary 4.3 gives the integer decomposition property. Every face inherits smoothness and the edge bound, so Theorem 1.2 gives condition (i) of Proposition 2.1. A proper face of dimension has edges at least ; Theorem 1.3, applied twice to that face, gives condition (ii). The exchange criterion completes the proof. □

References

[1]: C. Casagrande, Contractible classes in toric varieties, Mathematische Zeitschrift 243 (2003), 99–126. arXiv math/0111332.
[2]: A. Higashitani and H. Ohsugi, Toric ideals of Minkowski sums of unit simplices, Algebr. Comb. 3 (2020), 831–837. DOI 10.5802/alco.117.
[3]: L. Song and H. Wen, On Property of line bundles on smooth projective toric varieties, preprint, 2026. arXiv 2606.30160.

Quadratic generation for smooth polytopes with long edges