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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 $$P$$.

##### MSC:
 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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