  We fix the lexicographic order $\prec$ on the polynomial ring
$S=k[x_{1},...,x_{n}]$ over a ring $k$. We define $\Hi^{\prec\Delta}_{S/k}$,
the moduli space of reduced Gr\"obner bases with a given finite standard set
$\Delta$, and its open subscheme $\Hi^{\prec\Delta,\et}_{S/k}$, the moduli
space of families of $#\Delta$ points whose attached ideal has the standard set
$\Delta$. We determine the number of irreducible and connected components of
the latter scheme; we show that it is equidimensional over ${\rm Spec}\,k$; and
we determine its relative dimension over ${\rm Spec} k$. We show that analogous
statements do not hold for the scheme $\Hi^{\prec\Delta}_{S/k}$. Our results
prove a version of a conjecture by Bernd Sturmfels.
