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 talk, we consider $J$ as an ideal generated by 2-minors of a $2\times n$ matrix of linear forms and $I$ stands for the Jacobian ideal of $J$. We show that the pair $J\subseteq I$ is Aluffi torsion-free if and only if in the Kronecker-Weierstrass normal form of the matrix, there is no any Jordan block, or equivalently, $I_r(\Theta) = {\mathfrak m}^r$, where $r$ is codimension of $J$, $\Theta$ stands for the Jacobian matrix of $J$ and ${\mathfrak m}$ is the homogeneous maximal ideal of $k[{\bf X}]=k[x_1,\ldots,x_n]$. This motivates us to conjecture that, if $J\subset k[{\bf X}]$ is an ideal of codimension $r\geq 2$, generated by 2-forms, and if $I$ denotes the ideal generated by $r$-minors of the Jacobian matrix $\Theta$ of $J$, then $I$ is ${\mathfrak m}$-primary if and only if $I={\mathfrak m}^r$.

Also, we find conditions for edge ideal of a graph and its Jacobian ideal to be Aluffi torsion-free pair. In this regard, we give some necessary and sufficient conditions on graphs equivalent to the Aluffi torsion-free property.

References
[1] P. Aluffi, Shadows of blow-up algebras, Tohoku Math. J. (2) 56, no. 4 (2004)
593–619.
[2] A. Nasrollah Nejad, A. Simis, The Aluffi algebra, J. Singul. 3 (2011) 20–47.
[3] A. Nasrollah Nejad, R. Zaare-Nahandi, Aluffi torsion-free ideals, Journal
of Algebra 346 (2011) 284-298.
[4] R. Zaare-Nahandi, R. Zaare-Nahandi, Gr¨obner basis and free resolution of
the ideal of 2-minors of a 2 à— n matrix of linear forms, Comm. Algebra, 28,
no. 9 (2000) 4433–4453.