The punctual Hilbert scheme of 4 points in affine 3-space

Abstract: The $n$-th punctual Hilbert scheme $\operatorname{Hilb}^n_0(\mathbb{A}^d)$ of points of affine $d$-space parametrises ideals of finite co-length $n$ of the ring of functions on $d$-dimensional affine space, whose radical is the maximal ideal at the origin (equivalently, subschemes of length $n$ with support at the origin). A classical theorem of Briancon claims the irreducibility of this space for $d=2$ and arbitrary $n$. The case of a small number of points being straightforward, the first nontrivial case is the case of $4$ points in $3$-space. We show, answering a question of Sturmfels, that over the complex numbers $\operatorname{Hilb}^4_0(\mathbb{A}^3)$ is irreducible. We use a combination of arguments from computer algebra and representation theory.

commutative algebraalgebraic geometry

Audience: researchers in the topic

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