  We deal with the quasi-symmetric algebra introduced by Paolo Aluffi, here
named (embedded) Aluffi algebra. The algebra is a sort of "intermediate"
algebra between the symmetric algebra and the Rees algebra of an ideal, which
serves the purpose of introducing the characteristic cycle of a hypersurface in
intersection theory. The results described in the present paper have an
algebraic flavor and naturally connect with various themes of commutative
algebra, such as standard bases \'a la Hironaka, Artin--Rees like questions,
Valabrega--Valla ideals, ideals of linear type, relation type and analytic
spread. We give estimates for the dimension of the Aluffi algebra and show
that, pretty generally, the latter is equidimensional whenever the base ring is
a hypersurface ring. There is a converse to this under certain conditions that
essentially subsume the setup in Aluffi's theory, thus suggesting that this
algebra will not handle cases other than the singular locus of a hypersurface.
The torsion and the structure of the minimal primes of the algebra are
clarified. In the case of a projective hypersurface the results are more
precise and one is naturally led to look at families of projective plane
singular curves to understand how the property of being of linear type
deforms/specializes for the singular locus of a member. It is fairly elementary
to show that the singular locus of an irreducible curve of degree at most 3 is
of linear type. This is roundly false in degree larger than 4 and the picture
looks pretty wild as we point out by means of some families. Degree 4 is the
intriguing case. Here we are able to show that the singular locus of the
generic member of a family of rational quartics, fixing the singularity type,
is of linear type. We conjecture that every irreducible quartic has singular
locus of linear type.
