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The depth of the Rees algebra of three general binary forms. (English) Zbl 1423.13007
In the paper under review, the authors prove that the Rees algebra of an ideal generated by three general binary forms of same degree $$d \geq 5$$ has depth one. This result is in sharp contrast to recent akin statements regarding the depth of the Rees algebra of an ideal $$I$$ when $$I$$ is an almost complete intersection, (see [J. Hong et al., J. Symb. Comput. 43, No. 4, 275–292 (2008; Zbl 1139.13013); J. Commut. Algebra 5, No. 2, 231–267 (2013; Zbl 1274.13015); M. E. Rossi and I. Swanson Contemp. Math. 331, 313–328 (2003; Zbl 1089.13501); A. Simis and S. O. Tohǎneanu, Collect. Math. 66, No. 1, 1–31 (2015; Zbl 1329.13008)], where it has been proved that the Rees algebra is almost Cohen-Macaulay.
The main tools that are used in the proof are: the Ratliff-Rush filtration and the Huckaba-Marley test. The authors also give a conjecture that implies the main theorem of the paper.
MSC:
 13A02 Graded rings 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13C15 Dimension theory, depth, related commutative rings (catenary, etc.) 13D02 Syzygies, resolutions, complexes and commutative rings 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
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References:
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