## The Aluffi algebra

Abbas Nasrollah Nejad and Aron Simis

Journal of Singularities

volume 3 (2011), 20-47

Received: 3 January 2011. Received in revised form: 20 February 2011.

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Abstract:

We deal with the * quasi-symmetric algebra* introduced
by Paolo Aluffi, here named the (embedded) Aluffi algebra.
This 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 à 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.

Mathematical Subject Classification:

Primary 13A30, 14B05, 14D06, 14H10; Secondary 13B25, 13C15, 13D02, 13F45, 13P10, 14H50

Author(s) information:

Abbas Nasrollah Nejad | Aron Simis |

Departamento de Matemática, CCEN | Departamento de Matemática, CCEN |

Universidade Federal de Pernambuco | Universidade Federal de Pernambuco |

Cidade Universitária | Cidade Universitária |

50740-540 Recife, PE, Brazil | 50740-540 Recife, PE, Brazil |

email: abbasnn@dmat.ufpe.br | email: aron@dmat.ufpe.br |