
TL;DR
This paper explores the algebraic properties of the Aluffi algebra, an intermediate algebra between the symmetric and Rees algebras, focusing on its dimension, structure, and behavior in hypersurface singularities.
Contribution
It provides new estimates for the Aluffi algebra's dimension, characterizes its structure, and investigates its behavior in the context of hypersurface singularities and projective curves.
Findings
The Aluffi algebra is equidimensional for hypersurface rings.
Singular locus of low-degree curves (degree ≤ 3) is of linear type.
Degree 4 cases exhibit complex behavior, with conjectures on quartic curves.
Abstract
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…
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