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 \({\mathfrak {p}}= (x,y)\) is a height two prime ideal. This paper concerns the structure of \({\mathfrak {p}}\)-primary ideals in S. Huneke, Seceleanu, and the authors showed that for \(e \ge 3\), there are infinitely many pairwise non-isomorphic \({\mathfrak {p}}\)-primary ideals of multiplicity e. However, we show that for \(e \le 4\) there is a finite characterization of the linear, quadric and cubic generators of all such \({\mathfrak {p}}\)-primary ideals. We apply our results to improve bounds on the projective dimension of ideals generated by three cubic forms.