Quadratic generation of ideals defining nonsigular toric 3-folds

Abstract: Let $X$ be a projective line bundle over a nonsingular toric surface which is a blowup the projective plane along at most 4 invariant points, or a blowup the product of the projective lines along at most 4 invariant points. Let $L$ be an ample line bundle on $X$. Then $L$ defines a projectively normal embedding to big projective space. We show the ideal of this embedded $X$ is generated by elements of degree two.

• The paper proves that every ample embedding of specified smooth toric 3-folds has a defining ideal generated by quadratic binomials, confirming Sturmfels’ conjecture for these cases.
• The proof reduces cubic relations to quadrics using lattice-square rewriting, midpoint-parity analysis, and slice-wise combinatorics of toric polytopes.
• The result applies to toric \(\mathbb{P}^1\)-bundles over mildly blown-up \(\mathbb{P}^2\) or \(\mathbb{F}_0\), and extends to related 3-folds with at most four invariant blowups while leaving broader cases open.

Overview and context

This paper by Shoetsu Ogata addresses Sturmfels' 1995 conjecture that for a normally generated ample line bundle $L$ on a smooth toric variety $X$, the defining ideal $$I(X,L) = \ker(\mathrm{Sym}\,\Gamma(X,L) \to \bigoplus_k \Gamma(X, L^{\otimes k}))$$ of the induced projective embedding is generated in degree two. The main result gives an affirmative answer for a class of polarized nonsingular toric 3-folds: if $X$ is a toric $\mathbb{P}^1$-bundle over a smooth toric surface $Y$ that is either $\mathbb{P}^2$ blown up at at most 3 invariant points or $\mathbb{F}_0 = \mathbb{P}^1\times\mathbb{P}^1$ blown up at at most 4 invariant points, then $I(X,L)$ is generated by quadrics (binomials of degree two) for every ample line bundle $L$ (2608.19577).

The paper situates itself within an established chain of results. In dimension two, Koelman proved both normal generation of all ample line bundles and the quadratic-generation criterion for embedded toric surfaces. In higher dimensions, Demazure's theorem guarantees very ampleness of every ample line bundle on a smooth toric variety, but not projective normality: Ogata himself previously exhibited ample line bundles on singular toric 3-folds that are very ample yet fail to be normally generated. For smooth toric 3-folds admitting a surjective morphism to $X$0, or with $X$1 for the polarization, Ogata established normal generation. A prior weak version of the conjecture showed such $X$2 are zero-sets of quadratic binomials; the present paper strengthens this from set-theoretic quadratic definition to ideal-theoretic quadratic generation.

Toric dictionary

The paper works throughout via the standard polytope correspondence. An ample line bundle $X$3 on a projective toric variety corresponds to a lattice polytope $X$4 with $X$5, and normal generation is equivalent to the lattice-point identity $X$6. Smoothness is characterized by unimodularity of each vertex cone $X$7.

The relevant class of 3-fold polytopes arises from Ogata's earlier classification theorem: a polarized smooth toric 3-fold $X$8 with no interior lattice points in its polytope $X$9 (equivalently $$I(X,L) = \ker(\mathrm{Sym}\,\Gamma(X,L) \to \bigoplus_k \Gamma(X, L^{\otimes k}))$$0) must be one of:

1. a blowup of $$I(X,L) = \ker(\mathrm{Sym}\,\Gamma(X,L) \to \bigoplus_k \Gamma(X, L^{\otimes k}))$$1 at at most 4 invariant points,
2. a blowup of a $$I(X,L) = \ker(\mathrm{Sym}\,\Gamma(X,L) \to \bigoplus_k \Gamma(X, L^{\otimes k}))$$2-bundle over $$I(X,L) = \ker(\mathrm{Sym}\,\Gamma(X,L) \to \bigoplus_k \Gamma(X, L^{\otimes k}))$$3 at at most 2 invariant points, or
3. a $$I(X,L) = \ker(\mathrm{Sym}\,\Gamma(X,L) \to \bigoplus_k \Gamma(X, L^{\otimes k}))$$4-bundle over a smooth toric surface,

with all ample line bundles projectively normal. Case (3) covers the paper's targets, and the base polygon $$I(X,L) = \ker(\mathrm{Sym}\,\Gamma(X,L) \to \bigoplus_k \Gamma(X, L^{\otimes k}))$$5 takes one of two explicit forms: a truncated dilation of the basic simplex $$I(X,L) = \ker(\mathrm{Sym}\,\Gamma(X,L) \to \bigoplus_k \Gamma(X, L^{\otimes k}))$$6, or a doubly-truncated rectangle $$I(X,L) = \ker(\mathrm{Sym}\,\Gamma(X,L) \to \bigoplus_k \Gamma(X, L^{\otimes k}))$$7.

Method: deforming degree-three binomial relations

The proof strategy exploits the fact that, by results of Ogata and Sturmfels, for a normally generated smooth toric variety of dimension $$I(X,L) = \ker(\mathrm{Sym}\,\Gamma(X,L) \to \bigoplus_k \Gamma(X, L^{\otimes k}))$$8 the ideal is generated in degrees at most $$I(X,L) = \ker(\mathrm{Sym}\,\Gamma(X,L) \to \bigoplus_k \Gamma(X, L^{\otimes k}))$$9 — so here by binomials of degree at most three (using Eisenbud–Sturmfels' binomiality). It suffices to show every cubic relation $X$0 among lattice points can be decomposed using quadratic relations.

The core combinatorial input is a lemma stating that any pair relation $X$1 inside one of the polygons $X$2 can be rewritten via a lattice square (parallel transform of the unit square): there exists a lattice square $X$3 with $X$4 realizing the same sum. This reduces the analysis to configurations where pairs lie in unit squares.

The case analysis then proceeds geometrically. Since any lattice point of the prism-like polytope lies on one of the two parallel faces $X$5, a cubic relation distributes as $X$6 (or symmetrically), reducing to two points on $X$7 and one on $X$8. When the midpoint $X$9 is integral, the key observation is that the centroid coincidence forces the midpoints of $\mathbb{P}^1$0 and $\mathbb{P}^1$1 to have identical "type" among $\mathbb{P}^1$2, i.e., matching integrality patterns of coordinates. Each type admits an explicit deformation into two quadratic relations, including the delicate perpendicular case of type $\mathbb{P}^1$3, handled by translating segments even when the translated segment exits the polytope but its endpoints remain.

When $\mathbb{P}^1$4, a second application of the square lemma replaces the offending segment on $\mathbb{P}^1$5 by one with integral midpoint, and a midpoint-type comparison shows that the difference vector $\mathbb{P}^1$6 matches either $\mathbb{P}^1$7 or $\mathbb{P}^1$8, again yielding the required factorization through quadrics.

Extension to arbitrary polarizations

The passage from the anticanonical-type polarization ($\mathbb{P}^1$9) to an arbitrary ample $Y$0 uses the fibered structure directly. Writing the polytope $Y$1 of $Y$2 between planes $Y$3 and $Y$4, each cross-section $Y$5 has the same number of vertices as $Y$6 with correspondingly parallel edges — in particular each $Y$7 is a nonsingular polygon of the same form $Y$8 or $Y$9. Given a cubic relation, parity of $\mathbb{P}^2$0-coordinates lets one choose pairs $\mathbb{P}^2$1 and $\mathbb{P}^2$2 whose midpoints have integral height $\mathbb{P}^2$3; the relation forces the midpoint of $\mathbb{P}^2$4 onto an integer slice $\mathbb{P}^2$5 as well. Applying the square lemma within slices and rerunning the midpoint-type argument completes the reduction. As a corollary, cases (1) and (2) of the classification also satisfy quadratic generation, since their polytopes admit the same slicing description.

Limitations and open questions

The result remains partial relative to the full conjecture. The class covered consists of toric $\mathbb{P}^2$6-bundles over surfaces obtained from $\mathbb{P}^2$7 or $\mathbb{P}^2$8 by few invariant blowups, together with the low-blowup cases (1) and (2); the general smooth toric 3-fold, let alone higher dimensions, is untouched. The proof relies essentially on the rigidity of the polygon types $\mathbb{P}^2$9 and on the slice-wise nonsingularity of $\mathbb{F}_0 = \mathbb{P}^1\times\mathbb{P}^1$0, so it does not obviously extend to bases requiring more blowups, where these structural lemmas may fail. The paper also inherits the hypothesis-free nature of the statement only through prior normal-generation results for these specific geometries; whether the deformation technique adapts to toric varieties without a product structure is left open, as is the corresponding question in dimension four and above.

Conclusion

The paper verifies Sturmfels' quadratic-generation conjecture for a concrete class of smooth toric 3-folds — $\mathbb{F}_0 = \mathbb{P}^1\times\mathbb{P}^1$1-bundles over mildly blown-up $\mathbb{F}_0 = \mathbb{P}^1\times\mathbb{P}^1$2 and $\mathbb{F}_0 = \mathbb{P}^1\times\mathbb{P}^1$3 — by combining Ogata's classification of polarizations with vanishing adjoint sections, a lattice-square rewriting lemma, and a midpoint-parity case analysis that factors every cubic binomial through quadrics. The argument is entirely combinatorial and effective, and the corollary extends coverage to all smooth toric 3-folds appearing in the classification with at most four invariant blowups.