A finite classification of (x, y)-primary ideals of low multiplicity

Let S be a polynomial ring over an algebraically closed field k . Let x and y denote linearly independent linear forms in S so that p = ( x , y ) is a height two prime ideal. This paper concerns the structure of p -primary ideals in S . Huneke, Seceleanu, and the authors showed that for e ≥ 3 , there are infinitely many pairwise non-isomorphic p -primary ideals of multiplicity e . However, we show that for e ≤ 4 there is a finite characterization of the linear, quadric and cubic generators of all such p -primary ideals. We apply our results to improve bounds on the projective dimension of ideals generated by three cubic forms.