Equigenerated Gorenstein ideals of codimension 3: with a chapter on general forms - Lira, Dayane Santos de (2022)
This thesis deals with equigenerated Gorenstein ideals of nite colength in a standard graded ring R = k[x1; : : : ; xn] over an in nite eld k. We focus especially on such ideals of codimension 3, by looking at properties involving the Macaulay inverse system, the degree of socle, the reduction number, and the Cohen-Macaulayness of the associated Rees algebra. A special attention is devoted to the classical problem of general forms, as in the well-known conjecture of Fröberg. Our interest is to understand the sparsity of Gorenstein ideals generated by general forms. We conjecture that if I R is an ideal generated by a general set of r n+2 forms of degree d 2, then I is Gorenstein if and only if d = 2 and r = n+1 2 1. We prove this conjecture for n = 3 and one of its implications for arbitrary n. Another theme considered in this thesis is what we called the colon problem, a subject related to the presentation of a Gorenstein ideal as a link I = (xm1 ; : : : ; xmn ) : f, for a form f 2 R. If I has nite colength and linear resolution, we establish under what conditions the form f is uniquely determined, in addition to determining its degree. As we show, this problem is related to the so-called Newton dual introduced by Costa Simis and further studied by various recent authors.