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We give a Gr$\ddot{\text{o}}$bner basis for the ideal of 2-minors of a $2\times n$ matrix of linear forms. The minimal free resolution of such an ideal is obtained in [4] when the corresponding Kronecker-Weierstrass normal form has no nilpotent blocks. For the general case, using this result, the Gr$\ddot{\text{o}}$bner basis and the Eliahou-Kervaire resolution for stable monomial ideals, we obtain a free resolution with the expected regularity. For a specialization of the defining ideal of ordinary pinch points, as a special case
of these ideals, we provide a minimal free resolution explicitly in terms of certain Koszul complex.
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