A Seventeen-Dimensional Component of the Hilbert Scheme of Eighteen Points on an Integral Curve

We give an explicit integral projective curve whose Hilbert scheme of eighteen points has a rational component of dimension seventeen. This answers affirmatively Problem 20 in the 2010 AIM workshop list Components of Hilbert Schemes, even with the curve required to be integral and projective. The curve comes from the known numerical semigroup \(\langle13,14,15,16,17,18,21,23\rangle\) of Herzog–Kumashiro–Stamate. A five-generator ideal of colength eighteen is a canonical module; its full Hilbert tangent space is the seventeen-dimensional normalization quotient. A flat family of truncated unit translates identifies an open subset of the Hilbert scheme with \(\A^{17}\) . Two independent finite syzygy certificates verify the tangent calculation over every field. In characteristic zero, a separate application of Kass's theorem gives a component of dimension \(d-1\) for every \(d\ge18\) on the same curve. The semigroup, canonical-module identities and general moduli theorem are established prior work; the point is their explicit Hilbert-scheme application. No absolute priority or minimal-length claim is made.