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Aluffi torsion-free ideals

A special class of algebras which are intermediate between the symmetric and the Rees algebras of an ideal was introduced by P. Aluffi in 2004 to define characteristic cycle of a hypersurface parallel to conormal cycle in intersection theory. These algebras are recently investigated by A. Nasrollah Nejad and A. Simis who named them Aluffi algebras. For a pair of ideals $J\subseteq I$ of a commutative ring $R$, the Aluffi algebra of $I/J$ is called Aluffi torsion-free if it is isomorphic to the Rees algebra of $I/J$. In this paper, ideals generated by 2-minors of a $2\times n$ matrix of linear forms and also edge ideals of graphs are considered and some conditions are presented which are equivalent to Aluffi torsion-free property of them. Also many other examples and further questions are presented.