  Let $k$ be an algebraically-closed field, and $B$ a unital, associative
$k$-algebra with $n := \dim_kB < \infty$. For each $1 \le m \le n$, the
collection of all $m$-dimensional subalgebras of $B$ carries the structure of a
projective variety, which we call $\operatorname{AlgGr_m(B)}$. The group
$\operatorname{Aut}_k(B)$ of all $k$-algebra automorphisms of $B$ acts
regularly on $\operatorname{AlgGr}_m(B)$. In this paper, we study the problem
of explicitly describing $\operatorname{AlgGr}_m(B)$, and classifying its
$\operatorname{Aut}_k(B)$-orbits. Inspired by recent results on maximal
subalgebras of finite-dimensional algebras, we compute the homogeneous
vanishing ideal of $\operatorname{AlgGr}_{n-1}(B)$ when $B$ is basic, and
explictly describe its irreducible components. We show that in this case,
$\operatorname{AlgGr}_{n-1}(B)$ is a finite union of
$\operatorname{Aut}_k(B)$-orbits if $B$ is monomial or its Ext quiver is Schur,
but construct a class of examples to show that these conditions are not
necessary.
