Torsion-free Aluffi algebras

A pair of ideals J⊆I⊆R has been called Aluffi torsion-free if the Aluffi algebra of I/J is isomorphic to the corresponding Rees algebra. We give necessary and sufficient conditions for the Aluffi torsion-free property in terms of the first syzygy module of the form ideal J⁎ in the associated graded ring of I. For two pairs of ideals J1,J2⊆I such that J1−J2∈I2, we prove that if one pair is Aluffi torsion-free the other one is so if and only if the first syzygy modules of J1 and J2 have the same form ideals. We introduce the notion of strongly Aluffi torsion-free ideals and present some results on these ideals.