Volume Bounds for Holes in Very Ample Semigroups

Definitions and mathematical statement

Let $A$ be a finite set of lattice points, let $P=\operatorname{conv}(A)$, and form the homogenized affine semigroup \[S_A=\left\{\sum_{a\in A}n_a(a,1):n_a\in{\mathbb{Z}}_{\geq0}\right\}.\] Write $L_A=\operatorname{gp}(S_A)$ for its group of differences and $\overline S_A=\operatorname{cone}(S_A)\cap L_A$ for its saturation. A hole is an element of $\overline S_A\setminus S_A$; its height is its last coordinate, the homogeneous degree. In the usual semigroup meaning of very ample, these holes are finite in number. For this interpretation put \[h(A)=\max\{k:(x,k)\in\overline S_A\setminus S_A\},\] with $h(A)=0$ if there are no holes. The source assumes that the configuration gives a very ample embedding; its exact geometric/semigroup convention is not stated and must be confirmed. In particular $A$ need not equal $P\cap M$: the semigroup must be generated by the specified configuration, not silently by every lattice point of $P$. To specify one standard volume convention, choose $a_0\in A$, let $M_A=\operatorname{span}_{{\mathbb{Z}}}\{a-a_0:a\in A\}$ have rank $d$, and normalize volume in its real span so a fundamental parallelepiped of $M_A$ has volume one. Then $\operatorname{Vol}_{\mathrm{norm}}(P)=d!\operatorname{vol}_{M_A}(P-a_0)$. The question is whether the highest-hole height can be bounded in terms of this normalized volume, schematically \[h(A)\leq B\left(\operatorname{Vol}_{\mathrm{norm}}(P)\right)\] for a bound independent of the individual admissible configuration. The extract does not specify whether dimension is fixed or whether the ambient lattice rather than the configuration's generated lattice is used for normalization. These choices cannot be silently changed: using holes in a larger lattice may introduce elements absent from $L_A$ and change finiteness. The supplied 2026-08-21 triage labels this open and says general volume bounds for lattice polytopes do not settle the requested semigroup-height bound.

Short English statement

Ask whether the largest degree of a missing lattice-semigroup element is bounded by the normalized volume of the convex hull. The actual generating configuration matters: it may omit lattice points of that hull, even though it gives a very ample embedding.

Sources

• Local UnsolvedMath registry, permanent ID 30000819, source problem catalogue entry, Volume Bounds for Holes in Very Ample Semigroups, proposed year 2007; no individual proposer supplied.
• Supplied citation: Mini-Workshop: Projective Normality of Smooth Toric Varieties (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/39, pp. 2283--2320: https://doi.org/10.4171/owr/2007/39. Supplied literature label: original OWR report.
• Supplied literature label: OWR problem text, https://oa.tib.eu/renate/bitstreams/29b4c7a6-7c9e-42dd-8780-60d49fb0675c/download.