  Using the syzygy method, established in our earlier paper, we characterize
the combinatorial stratification of the variety of two-dimensional real generic
algebras. We show that there exist exactly three different homotopic types of
such algebras and relate this result to potential applications and known facts
from qualitative theory of quadratic ODEs. The genericity condition is crucial.
For example, the idempotent geometry in Clifford algebras or Jordan algebras of
Clifford type is very different: such algebras always contain nontrivial
submanifolds of idempotents.
