The Hilbert-basis-hole conjecture for Kreuzer–Skarke Calabi–Yau threefolds

Let $X$ be a smooth Calabi–Yau threefold arising from the Kreuzer–Skarke database, and let a non-trivial Hilbert basis element be a divisor class in the Hilbert basis under consideration. A hole is a divisor class in the relevant effective semigroup or cone that is not itself effective. Hilbert-basis-hole conjecture. All non-trivial Hilbert basis elements on smooth Calabi–Yau threefolds arising from the Kreuzer–Skarke database are holes. The conjecture is based on empirical checks for Calabi–Yau threefolds with $h^{1,1}c=6$, including both big and non-big non-trivial Hilbert basis elements. It remains open beyond the computed data, and the source notes that the database evidence is strong rather than a proof.