The depth of the Rees algebra of three general binary forms

Abstract

One proves that the Rees algebra of an ideal generated by three general binary forms of same degree ≥ 5 has depth one. The proof hinges on the behavior of the Ratliff-Rush filtration for low powers of the ideal and on establishing that certain large matrices whose entries are quadratic forms have maximal rank. One also conjectures a shorter result that implies the main theorem of the paper.2010 Mathematics Subject Classification. 13A02, 13A30, 13C15, 13D02, 13D40.Plugging this back into the above equation, after canceling P x d , and after simplifying by y, one has P y 2 g 1 + Q x d−1 + y 2 Q g 2 − xyP g 2 + Rx d−2 + yRg 3 = 0.Again, plugging this back, canceling Q x d−1 , and simplifying by y yields P (yg 1 − xg 2 ) + Q (yg 2 − xg 3 ) + R (x d−2 + yg 3 ) = 0.