Unimodular Coverings of Adjoint Toric Polytopes

Definitions and mathematical statement

Let $X$ be a smooth projective toric variety of dimension $d$, let $A$ be ample and $B$ nef, and retain the intended adjoint divisor $D$. The supplied clean reconstruction proposes \[ D_{\mathrm{cand}}=(d+2)A+B+K_X, \] whereas the original/statement fields print $D=A^{d+2}+B+K_X$. The verification explicitly says the OCR expression could not be recovered unambiguously and assigns reconstructed-unverified status. Tensor-power notation for a line bundle could explain an additive multiple, but that interpretation is not certified by the record. The candidate is therefore conditional, not promoted to a recovered divisor formula.

For a specified integral torus-invariant Cartier representative $D=\sum_\rho a_\rho D_\rho$, with character lattice $M$ and primitive ray generators $v_\rho$, its polytope convention is \[ P_D=\{u\in M\otimes_{\mathbb{Z}}{\mathbb{R}}:\langle u,v_\rho\rangle\geq-a_\rho\text{ for all }\rho\}. \] The source presupposes an associated lattice polytope but does not give this representative or confirm the divisor expression. For a lattice polytope $P$, normality here means the integer-decomposition property \[ mP\cap M=\underbrace{(P\cap M)+\cdots+(P\cap M)}_{m\text{ summands}} \quad\text{for every integer }m\geq1. \]

A unimodular covering means a finite collection of lattice simplices $\Delta_i\subseteq P$ whose union is $P$, each unimodular in the relevant lattice. For a full-dimensional $d$-simplex with vertices $u_0,\ldots,u_d$, unimodular means that $u_1-u_0,\ldots,u_d-u_0$ form a ${\mathbb{Z}}$-basis of $M$, equivalently normalized volume one. If $P$ has smaller dimension, the affine-lattice convention must be stated. A covering need not be a face-to-face triangulation, regular triangulation or disjoint-interior decomposition, and no such stronger conclusion is substituted.

For the intended adjoint family, assuming $P_D$ normal, the question is whether $P_D$ has a unimodular covering. It does not ask whether every arbitrary normal lattice polytope does. The original attributes normality to a cited vanishing theorem, while the clean version takes it as a hypothesis; the theorem and its assumptions are not reproduced, so normality is not independently asserted.

The 2026-08-21 literature assessment says the divisor/polytope was omitted and no general answer located. The current reconstruction supplies a tentative expression but the verification still withholds confirmation. The open label is retained as dated metadata, not a verified current theorem status.

Short English statement

For the source's intended normal adjoint toric polytope, ask whether it can be covered by unimodular lattice simplices. A covering is weaker than a triangulation. The divisor expression is explicitly reconstructed and unverified, so the cleaned multiple $(d+2)A$ remains a candidate rather than a silently accepted OCR repair.

Sources

• Local UnsolvedMath registry, permanent ID 30000814, source code catalogue entry, "Unimodular Coverings of Adjoint Toric Polytopes," including the original $A^{d+2}+B+K_X$ expression, candidate cleaned formula and reconstructed-unverified warning.
• Mini-Workshop: Projective Normality of Smooth Toric Varieties (2008), Oberwolfach Reports, pp. 2283--2320, DOI 10.4171/owr/2007/39: https://doi.org/10.4171/owr/2007/39. This is the sole supplied literature reference. The citation uses 2008, while the DOI and proposed-year metadata use 2007.
• The normality claim refers to a vanishing theorem not identified in the supplied record; no separate citation is invented.
• Literature triage checked 2026-08-21; record updated 2026-09-16. No external source has been visited or independently checked here.