I would be very grateful for any comment or a reference on the following question.

Let $Hilb_0^n({\mathbb C}^2)$ be the Hilbert scheme of n points in ${\mathbb C}^2$ concentrated, set theoretically, at the origin. Let $X$ be the locus of $Hilb_0^n({\mathbb C}^2)$ formed by curvilinear ideals.

Is the complement of $X$ in $Hilb_0^n({\mathbb C}^2)$ a divisor or it has codimension > 1 ?