  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.
