Generic Multiplicities of Hilbert Schemes on Thickened Lines

Let \(T_{n,\beta}\) be the Hilbert scheme of \(n\) points on the thickened line defined by \(y^\beta=0\) over \(\mathbb C\). For the component indexed by a partition \(\lambda\) of \(n\) with largest part at most \(\beta\), we compute its ambient generic multiplicity as \(\prod_i\binom{\beta}{\lambda_i}\). The proof uses disjoint-support factorization and a cyclic chart in which a monic polynomial divides \(t^\beta\); the remainder equations form a weighted complete intersection.

In particular, a chart of \(T_{2,3}\) has generic local ring \(K[u]/(u^3)\), whereas imposing a repeated-root discriminant produces \(K[u]/(u^2)\). This distinguishes ambient multiplicity from the multiplicity of a support stratum. We compare with the known multinomial multiplicity of the characteristic-polynomial nilpotent fiber, and describe reduced Jordan strata and their closure obstruction.

The stated geometric theorem is complete; the broad AIM categorical cell question remains unresolved. This preprint is unrefereed and makes no absolute-priority or formal-verification claim.