Fine Polyhedral Adjunction Theory

• Fine Polyhedral Adjunction Theory is a convex-geometric framework that extends classical adjunction principles to lattice polytopes using a refined lattice-sensitive construction.
• It yields improved decomposition results and strong finiteness theorems, sharpening combinatorial criteria through the use of Fine adjoint polytopes and enhanced spectral analysis.
• The theory bridges algebraic, toric, and symplectic geometry by connecting canonical forms on positive geometries with adjunction-type inequalities and mirror symmetry insights.

Fine Polyhedral Adjunction Theory is a convex-geometric framework that extends classical adjunction principles in algebraic geometry to lattice polytopes. It sharpens the original polyhedral adjunction theory by using a lattice-sensitive construction—the Fine adjoint polytope—which captures more refined structural and spectral properties of polytopes and their links to toric and symplectic geometry. This approach leads to improved decomposition results, strong finiteness theorems, and natural analogues of spectrum conjectures for toric varieties and beyond.

1. Definitions and Fundamental Objects

Given a full-dimensional rational polytope $P \subset \mathbb{R}^n$, a valid inequality is an affine functional $f(x) = \langle a, x \rangle - b$, with $a \in (\mathbb{Z}^n)^*$ primitive and $b \in \mathbb{Q}$, such that $P \subset \{x : f(x) \geq 0\}$. The Fine distance function for $a \in (\mathbb{R}^n)^*$ is $d_P^F(a) := \min_{x \in P} \langle a, x \rangle$.

For real $s > 0$, the Fine adjoint polytope is

$$P^{F(s)} := \{x \in \mathbb{R}^n : \langle a, x \rangle \geq d_P^F(a) + s,\ \forall\, 0 \neq a \in (\mathbb{Z}^n)^*\}.$$

The Fine interior or "heart," $P^F := P^{F(1)}$, plays a central role in adjunction questions.

Only finitely many relevant directions $a$ (relevant normals) are needed to determine $P^{F(s)}$, specifically those primitive elements in the convex hull of the minimal facet normals of $P$.

The Fine Q-codegree is defined as:

$$\mu^F(P) := \left(\sup\{s > 0 : P^{F(s)} \neq \emptyset\}\right)^{-1},$$

where the supremum is realized and the associated Fine core is $core^F(P) := P^{F(s_0)}$, $s_0 = 1/\mu^F(P)$ (Mora et al., 2023, Mora et al., 6 Jan 2026).

2. Structural Results and Decomposition Theorems

Fine Polyhedral Adjunction Theory recovers and sharpens several structural theorems about lattice polytopes, particularly concerning decomposition into Cayley sums. The Cayley sum of polytopes $P_0, \ldots, P_t \subset \mathbb{R}^k$ is:

$$P_0 * \cdots * P_t := \text{conv}\left\{ (P_0 \times 0), (P_1 \times e_1), \ldots, (P_t \times e_t) \right\} \subset \mathbb{R}^k \times \mathbb{R}^t.$$

The Fine decomposition bound is given by

$$d^F(P) = \begin{cases} 2(n - \lfloor \mu^F(P) \rfloor), & \mu^F(P) \notin \mathbb{Z}, \ 2(n - \mu^F(P)) + 1, & \mu^F(P) \in \mathbb{Z}. \end{cases}$$

Fine Decomposition Theorem: If $P$ is a lattice polytope in $\mathbb{R}^n$ not equivalent to the standard simplex $\Delta_n$ and $n > d^F(P)$, then $P$ splits nontrivially as a Cayley sum of polytopes of dimension at most $d^F(P)$ (Mora et al., 2023).

The proof strategy exploits the natural projection to the core, controlled by $\mu^F$, to exhibit a lattice hyperplane splitting and to iteratively peel off Cayley factors. This process yields sharper combinatorial criteria for decomposability than classical adjunction, with the strict inequality $\mu(P) \leq \mu^F(P)$ playing a critical role.

3. Fine Spectra, Finiteness, and Classification

For each $n \in \mathbb{N}$ and $\epsilon > 0$, the Fine spectrum is defined as

$$S^n_F(\epsilon) := \{\mu^F(P) : P \text{ an } n\text{-dimensional lattice polytope},\ \mu^F(P) \geq \epsilon\}.$$

A fundamental result is finiteness of the Fine spectrum: For fixed $n$ and $\epsilon > 0$, $S^n_F(\epsilon)$ is finite. No $\alpha$-canonical assumption is needed, unlike in classical theory (Mora et al., 2023, Mora et al., 6 Jan 2026).

Classification of maximal spectrum values (Mora et al., 6 Jan 2026):

Value for $\mu^F(P)$	Polytope Description
$d+1$	Standard unimodular simplex $\Delta_d$
$d$	Lattice-preserving projection onto $\Delta_{d-1}$ (Lawrence prism, Cayley of segments)
$d - 1/2$	(Unique) "exceptional simplex," a $(d-2)$-fold pyramid over $2\cdot \Delta_2$
$d-1$	Projection onto $\Delta_{d-2}$ not lying in above classes

For small dimensions, explicit spectra are available:

• $d=1: S^F_1 = \{2/k : k \in \mathbb{N}\}$
• $d=2: S^F_2 = \{2/k, 3/k : k\in\mathbb{N}\}$
• $$d=3: S^F_3 = \{q/(k\ell) : q\in\{2,3,4,5,7,11,13,17,19\}, k,\ell \in \mathbb{N} \}$$

Computational techniques include the "Fine mountain" polytope (a parameterized polytope in $\mathbb{R}^d \times \mathbb{R}$), vertex enumeration, and mixed-integer linear programming (MILP) to enumerate possible core-normal configurations.

4. Comparison with Classical Polyhedral Adjunction Theory

Classical adjunction theory associates to $P$ the adjoint polytope

$$P^{(s)} = \{x \in P : \langle a_i, x \rangle \geq b_i + s,\ \forall\text{ facet normals } a_i\},$$

considering only the normals that actually define the facets. Fine Polyhedral Adjunction Theory, in contrast, takes as valid all primitive lattice functionals admitting $P \subset \{x : \langle a, x\rangle \geq b\}$, reflecting the full lattice geometry.

Consequences:

• Fine adjoints $P^{F(s)}$ are generally smaller ($P^{F(s)} \subseteq P^{(s)}$), and $\mu(P) \leq \mu^F(P)$.
• The monotonicity property $P \subset Q \implies \mu^F(P) \geq \mu^F(Q)$ holds for $\mu^F$, but not for classical $\mu$.
• The spectrum finiteness result in Fine theory holds without assumptions on the normal fan, whereas the classical spectrum requires the fan to be $\alpha$-canonical.

Certain polytopes have disjoint classical and Fine cores, illustrating the strict refinement (Mora et al., 2023).

5. Canonical Forms and the Role of Adjoints

Fine Polyhedral Adjunction Theory interfaces with the construction of canonical forms on positive geometries, particularly for polytopes arising from the projectivization of polyhedral cones. For a pointed, full-dimensional cone $C \subset \mathbb{R}^{m+1}$, the canonical form in homogeneous coordinates is

$$\Omega_C(x) = \frac{A(x)}{\prod_{F \in f(C)} L_F(x)}\, dx,$$

where $B(x) = \prod_{F} L_F(x)$ encodes the facets and $A(x)$ is Warren's adjoint polynomial of the dual cone $C^{\vee}$ (Gaetz, 9 Apr 2025).

The adjoint $A(x)=\mathrm{adj}_{C}(x)$ is constructed so as to vanish along the "residual arrangement," i.e., those intersections of supporting hyperplanes that do not correspond to actual faces, thereby cancelling all unwanted poles outside the polytope. It is characterized by the homogeneity condition $\deg(A) = |f(C)| - \dim(C) - 1$ and a uniqueness property as a vanishing polynomial for all residual loci (Gaetz, 9 Apr 2025).

A canonical example: For the standard cube in $\mathbb{R}^3$, the canonical form is

$$\Omega_P(x) = \frac{x_1^2 + x_2^2 + x_3^2 - x_0^2}{\prod_{i=1}^3 (x_0 - x_i)(x_0 + x_i)} dx_1\,dx_2\,dx_3,$$

which has simple poles only on the six cube faces and no other poles in the interior.

6. Connections to Geometry and Topology

Fine Polyhedral Adjunction Theory has implications in the study of adjunction-type inequalities in almost-complex and symplectic geometry. In four-manifolds, the theory supports the extraction of classical adjunction genus bounds by decomposing the space into "polyhedral" strata—small pieces each carrying a taming symplectic form—and applying sectorwise inequalities such as the slice–Bennequin inequality (Lambert-Cole, 2021). This extends the adjunction principle beyond settings requiring a global symplectic form or Seiberg–Witten invariants, highlighting the flexibility and reach of the polyhedral approach.

7. Open Problems and Future Directions

Open questions include the precise structure and possible accumulation points of the lower part of the Fine spectrum (in particular, for large codimension in high dimensions); the classification of possible fine-core normal configurations in general; and the fine-tuning of canonical forms for polytopes outside the simplicial or reflexive cases (Mora et al., 6 Jan 2026).

Additionally, the connection between Fine adjunction invariants and global geometric or topological invariants continues to provide a promising avenue for further development, particularly at the interface with combinatorial and algebraic aspects of mirror symmetry, positive geometry, and log-canonical models.