Non-Smoothable Nestings of Fat Points

• The paper establishes explicit lower bounds and systematic construction methods showing that nested fat point configurations are generally non-smoothable in higher dimensions.
• It employs nested Hilbert scheme theory alongside deformation analysis, including graded Betti numbers and Białynicki–Birula decomposition, to investigate smoothability.
• These findings have significant implications for birational geometry, component stratification, and the reducibility of Hilbert schemes on singular hypersurfaces.

A non-smoothable nesting of fat points refers to an inclusion of two or more supported, zero-dimensional subschemes ("fat points") of a smooth quasi-projective variety $X$, such that the nested configuration does not lie in the closure of the locus of reduced schemes (i.e., it cannot be approximated by configurations of distinct points). This phenomenon is rigorously captured by the theory of nested Hilbert schemes, which for a non-decreasing sequence $\underline d=(d_1,\dots,d_r)$, parameterizes chains

$$Z^{(1)}\subset Z^{(2)}\subset\cdots\subset Z^{(r)}\subset X\times B$$

of closed subschemes, each flat over $B$ of relative length $d_i$. The existence, structure, and systematic construction of non-smoothable nestings of fat points in higher-dimensional ambient varieties reveal profound deviations from classical surface or curve cases and have major implications for the birational geometry and component structure of Hilbert schemes (Graffeo et al., 23 Jan 2026).

1. Structure and Smoothability of Nested Hilbert Schemes

A key foundational aspect is the precise definition of smoothability within the context of nested Hilbert schemes. For $r=2$, a point of $\Hilb^{d_1,d_2}(X)$ corresponds to an inclusion $[Z_1\subset Z_2]$ where each $Z_i$ is a fat point of length $d_i$. This nesting is termed smoothable if it lies in the closure of the locus where $\underline d=(d_1,\dots,d_r)$0 is reduced (i.e., a disjoint union of $\underline d=(d_1,\dots,d_r)$1 distinct points). The locus of non-smoothable nestings, conversely, consists of chains of fat points that cannot be realized as a limit of nestings of distinct reduced points. These distinctions become increasingly pronounced as the ambient dimension $\underline d=(d_1,\dots,d_r)$2 grows.

2. Lower Bounds for Non-Smoothable Nestings (Theorem A)

Consider $\underline d=(d_1,\dots,d_r)$3 with $\underline d=(d_1,\dots,d_r)$4 and an integer $\underline d=(d_1,\dots,d_r)$5. The construction utilizes two $\underline d=(d_1,\dots,d_r)$6-primary ideals: $\underline d=(d_1,\dots,d_r)$7 where $\underline d=(d_1,\dots,d_r)$8 is a general linear subspace of codimension 2, and $\underline d=(d_1,\dots,d_r)$9. The corresponding nesting $$Z^{(1)}\subset Z^{(2)}\subset\cdots\subset Z^{(r)}\subset X\times B$$0 defines a point in $$Z^{(1)}\subset Z^{(2)}\subset\cdots\subset Z^{(r)}\subset X\times B$$1. Via dimension computations:

• The expected dimension of the smoothable component is $$Z^{(1)}\subset Z^{(2)}\subset\cdots\subset Z^{(r)}\subset X\times B$$2.
• The Białynicki–Birula decomposition reveals that nestings of a specific Hilbert–Samuel type occupy a stratum of dimension $$Z^{(1)}\subset Z^{(2)}\subset\cdots\subset Z^{(r)}\subset X\times B$$3.

The gap

$$Z^{(1)}\subset Z^{(2)}\subset\cdots\subset Z^{(r)}\subset X\times B$$4

is non-positive precisely for $$Z^{(1)}\subset Z^{(2)}\subset\cdots\subset Z^{(r)}\subset X\times B$$5 (for $$Z^{(1)}\subset Z^{(2)}\subset\cdots\subset Z^{(r)}\subset X\times B$$6). Therefore, generic chains of this type do not lie in the smoothable locus, demonstrating explicit lower bounds for the existence of non-smoothable nestings of fat points in higher dimensions. The proof incorporates the semicontinuity of graded Betti numbers and tangent space analysis, ensuring correct Hilbert functions and component structure.

3. Systematic Construction of Non-Reduced Elementary Components (Theorem B)

A robust method exists for systematically producing generically non-reduced elementary components from reduced ones. Given $$Z^{(1)}\subset Z^{(2)}\subset\cdots\subset Z^{(r)}\subset X\times B$$7 as a generically reduced elementary component (the general nested chain has TNT: Trivial Negative Tangents), one can insert a layer $$Z^{(1)}\subset Z^{(2)}\subset\cdots\subset Z^{(r)}\subset X\times B$$8 between two ideals $$Z^{(1)}\subset Z^{(2)}\subset\cdots\subset Z^{(r)}\subset X\times B$$9 satisfying $B$0. Setting $B$1, the multi-length sequence becomes $B$2. The closure $B$3 retains the reduced structure but features increased negative tangent dimension, rendering it generically non-reduced.

Deformation theory furnishes the parameter count for the new moduli: $B$4 which is strictly positive given nontrivial sandwiching. In $B$5 with $B$6, a concrete example involves

$B$7

with TNT realized on the reduced component. Inserting $B$8 increases negative tangent dimension by $B$9 in $d_i$0, verifying the generic non-reducedness.

4. Reducibility of Hilbert Schemes on Singular Hypersurfaces (Theorem C)

The phenomenon extends to Hilbert schemes of points supported on singular hypersurfaces within $d_i$1. For a hypersurface $d_i$2 with a singular point of multiplicity at least 5, one shows explicitly that fat points of Hilbert function $d_i$3 (with total length 22) are forced to lie on $d_i$4. The corresponding local 2-step stratum $d_i$5 has dimension matching the smoothable component ($d_i$6), producing a new irreducible component—the 2-step locus. Consequently, $d_i$7 is reducible. Earlier results by Iarrobino establish reducibility of $d_i$8 for multiplicity $d_i$9.

5. Broader Consequences and Infinitely Many Non-Smoothable Chains

The existence of infinite families of non-smoothable chains of fat points in any dimension $r=2$0 is established. The outlined systematic construction offers a practical “toolbox” for generating non-reduced elementary components from reduced ones, broadening the catalog of known components. On singular threefolds or surfaces, new reducibility phenomena emerge that are tied to local singularity multiplicities. The global picture for nested Hilbert schemes in higher dimension is marked by wild birational geometry, a proliferation of elementary components (both reduced and non-reduced), and subtle wall-crossing behavior under deformations—qualities that significantly distinguish these moduli spaces from their curve or surface analogues (Graffeo et al., 23 Jan 2026).

6. Component Structure and Deformation Theory

Component structure in nested Hilbert schemes is decoded via decomposition techniques such as the Białynicki–Birula analysis for $r=2$1-actions and deformation-theoretic studies of tangent spaces. Semicontinuity of graded Betti numbers and TNT conditions enable rigorous identification of elementary components, distinguishing between reduced and non-reduced loci of nestings. The dimension formulae and moduli parameter counts allow for explicit predictions of component stratification and smoothability, underpinning further study of nested Hilbert scheme geometry.