Quadratically presented grade three Gorenstein ideals

Abstract

Let J be a quadratically presented grade three Gorenstein ideal in the standard graded polynomial ring R=k[x,y,z], where k is a field. Assume that R/J satisfies the weak Lefschetz property. We give the presentation matrix for J in terms of the coefficients of a Macaulay inverse system for J. (This presentation matrix is an alternating matrix and J is generated by the maximal order Pfaffians of the presentation matrix.) Our formulas are computer friendly; they involve only matrix multiplication; they do not involve multilinear algebra or complicated summations. As an application, we give the presentation matrix for J1=(xn+1,yn+1,zn+1):(x+y+z)n+1, when n is even and the characteristic of k is zero. Generators for J1 had been identified previously; but the presentation matrix for J1 had not previously been known. The first step in our proof is to give improved formulas for the presentation matrix of a linearly presented grade three Gorenstein ideal I in terms of the coefficients of the Macaulay inverse system for I.