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Polyhedral adjunction theory. (English) Zbl 1333.14010
Let \(P\subseteq \mathbb{R}^n\) be an \(n\)-dimensional rational polytope given by the inequalities \[ \left< a_i, \cdot\right> \geq b_i,\;\text{for}\;i=1,\ldots,m, \] where \(b_i\in \mathbb{Q}\) and \(a_i\in \left( \mathbb{Z}^n\right)^*\) are primitive. For \(x\in \mathbb{R}^n\), the lattice distance from the facets \(F_i\) of \(P\) is defined by \[ d_{F_i}(x)=\left<a_i,x\right>-b_i \] and the lattice distance with respect to \(\partial P\) by \[ d_P(x)=\min_{i=1,\ldots,m}d_{F_i}(x). \] The adjoint polytope of \(P\), denoted by \(P^{(s)}\), is defined as the set of those points in \(P\), whose lattice distance to every facet of \(P\) is at least \(s\). In order to study such polytopes, the authors study the following invariants. The \(\mathbb{Q}\)-codegree of a lattice polytope \(P\), is defined as \[ \mu(P)=\left( \sup\{s>0\;:\;P^{(s)}\neq \emptyset\}\right)^{-1}. \] For a rational polytope \(P\), its codegree is defined as \[ \text{cd}(P)=\min\{k\in\mathbb{N}\;:\;\text{int}(kP)\cap \mathbb{Z}^n \neq \emptyset\}. \] Also, the nef value of a rational polytope \(P\), is given as \[ \tau(P)=\left( \sup\{s>0\;:\;N(P^{s})=N(P)\}\right)^{-1}, \] which measure the positivity of the adjoint systems.
A. Dickenstein and B. Nill [Math. Res. Lett. 17, No. 3, 435–448 (2010; Zbl 1243.52010)] conjectured that if an \(n\)-dimensional lattice polytope \(P\) satisfies the inequality \(\text{cd}(P)>(n+2)/{2}\), then \(P\) is decomposed as a Cayley sum of lattice polytopes of dimension at most \(2(n+1-\text{cd}(P))\). A small different conjecture, comes from the replacement of the invariant \(\text{cd}(P)\), by the invariant \(\mu(P)\) and it is that for an \(n\)-dimensional lattice polytope \(P\) which satisfies the inequality \(\mu(P)>(n+1)/{2}\), then \(P\) decomposes as a Cayley sum of lattice polytopes of dimension at most \(\lfloor 2(n+1-\mu(P)) \rfloor\).
In their main result, the authors prove a slightly weaker version of the last conjecture, proving the conjecture for \(\mu(P)\geq (n+2)/2\). By this result, they generalize a result of [C. Haase et al., J. Reine Angew. Math. 637, 207–216 (2009; Zbl 1185.52012)], in which the conjecture is proved in the case of Gorenstein polytopes. In order to achieve this, the authors have as a toolbox, a lot of results by them, about the invariants \(\mathbb{Q}\)-codegree and the nef value of a rational polytope, under the natural projections \[ \pi_P : \mathbb{R}^n\longrightarrow \mathbb{R}^n/K(P), \] associated with \(P\), where \(K(P)\) is the linear space parallel to \(\text{aff}\left( P^{(1/\mu(P)}\right)\).
The authors also make a connection between all of the above in Ehrhart theory and on polarized toric varieties with dual defects, by proving for a lattice polytope \(P\) with \(\mu(P)\geq (3n+4)/4\) if \(\mu(P)\notin \mathbb{N}\) or \(\mu(P)\geq (3n+3)/4\) if \(\mu(P)\in \mathbb{N}\), that \(X_A\) has dual defect, where \(X_A\) be the toric variety embedded in \(\mathbb{P}^{\mid A\mid-1},\) where \(A\) is the set of lattice points of \(P\).

14C20 Divisors, linear systems, invertible sheaves
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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