Fixed point distribution on Hilbert scheme of points

Abstract: Let $\mathbf{k}$ be a closed field of characteristic zero. We prove that all monomial ideals sit in the curvilinear component of the Hilbert scheme of points of the affine space $\mathbb{A}_{\mathbf{k}}^n$, answering a long-standing question about the distribution of torus-fixed points among punctual components. This result confirms that the punctual Hilbert scheme is connected, a property previously established only for the full Hilbert scheme in 1966 by Hartshorne.