Research/Proposal A
Research proposal

When finite schemes deform

Artinian algebras, tangent spaces, and second-order obstructions

yx1zz²xy{1, x, y, z, z²}H = (1, 3, 1)

Question & scope

For Iₜ below, which embedded first-order deformations of the length-five quotient lift to second order?

Work over Q in Hilb⁵(A³), allowing all embedded deformations. The common graded Hilbert function describes the starting points, not a condition imposed on every deformation. For homogeneous, degree-zero deformations the two displayed seeds instead have tangent dimensions 5 and 5.

First calculation: a fixed-basis family

It=(x2,xy,xz,yz,y2−tz2,z3),B=(1,x,y,z,z2).I_t=(x^2,xy,xz,yz,y^2-tz^2,z^3),\qquad\mathcal B=(1,x,y,z,z^2).

Use the multiplication matrices in the displayed basis over Q[t]. At t=0 this is I₁. At t=−1, u=y+z and v=y−z turn the two quadratic relations y²+z² and yz into u² and v²; this identifies the fiber with I₂ over Q. This does not identify every nonzero rational fiber with I₂.

First verify that reduction gives this free rank-five Q[t]-basis. Keep the following matrix columns fixed: eₘ denotes the coordinate vector for the displayed basis monomial m, and perturbations in these columns vanish at every order. Only then linearize pairwise commutation. For a selected first-order direction Aᵢ, solve the order-ε² equations for Bᵢ in Mᵢ+εAᵢ+ε²Bᵢ. Report an actual lift or the exact unsatisfied linear compatibility condition.

Mxe1=ex,Mye1=ey,Mze1=ez,Mzez=ez2.\begin{aligned}M_xe_1&=e_x,&M_ye_1&=e_y,\\M_ze_1&=e_z,&M_ze_z&=e_{z^2}.\end{aligned}

Known boundaries

The 21/15 pair and three-variable monomial smoothness criteria are already in the cited literature. The pilot tests a specific deformation calculation. Before extending it, compare its equations and obstruction interpretation with the published criteria; reproducing a known local model is learning groundwork.

Three-variable monomial smoothness already has published criteria. A new contribution would require a clearly delimited family or deformation problem and a separate literature check.

The same Hilbert function, two different tangent spaces

S = ℚ[x, y, z]
I₁ = (x², xy, xz, y², yz, z³)
I₂ = (x², xy, xz, y², z²)
H = (1, 3, 1)   ·   dimℚ HomS(I₁, S/I₁) = 21   ·   dimℚ HomS(I₂, S/I₂) = 15

A classical pair of examples from the cited paper. Exact calculations were prepared with an assistant while scoping the project; these are not claimed as original results.

Supporting study notes

Classical background and worked examples for this proposal.

Work to undertake

  1. Verify the Q[t]-basis and compute multiplication matrices for Iₜ.
  2. Compute full embedded tangent spaces at t=0 and −1; keep degree-zero Hom separate.
  3. Solve a selected direction to order ε² and identify exactly where a lift succeeds or fails.
Commutative algebraAlgebraic geometryDeformation theory
  • Artinian algebras and socles
  • Syzygies and Hom spaces
  • Flat families
  • Artinian parameter rings

References

Jelisiejew, Ramkumar & Sammartano · The Hilbert scheme of points on a threefold2026 version
Giovenzana et al. · New components of Hilbert schemes of points and 2-step ideals2025

Books behind the question

Introduction to Commutative AlgebraEight Lectures on Monomial IdealsAlgebraic GeometryBasic Category Theory