Examples of Gorenstein domains and symbolic powers of monomial space curves (Q1175535)

Let \(C\) be a projective monomial space curve, \(C\subset\mathbb{P}^ 3_ k\), and let \(P\) be the ideal of \(C\) in \(A=k[x,y,z,w]\), \(k\) an arbitrary field. Many authors have studied ring conditions in order to obtain geometric properties of \(C\). In this paper the following main result is proved: if \(C\) is parametrically given by \(x=s^{a+b}\), \(y=s^ bt^ a\), \(z=s^ at^ b\), \(w=t^{a+b}\) with \(1\leq a<b\) and \(\text{gcd}(a,b)=1\) then the Rees ring of \(P\), \(r(P)=\oplus P^ nt^ n\), is a normal Gorenstein domain and the form ring of \(P\), \(G(P)=\oplus P^ n/P^{n+1}\), is a Gorenstein domain. Moreover, in the same case, one has \(P^ n=\overline{P^ n}=P^{(n)}\) for all \(n\geq 1\), where \(P^{(n)}\) is the \(n\)-th symbolic power of \(P\). In the second part of the paper the symbolic Rees ring of \(P\), \(S(P)=\oplus P^{(n)}t^ n\), is considered. It is proved that the property whether \(S(P)\) is an \(A\)-algebra of finite type does not depend upon the perfectness of \(P\). More precisely, in the theorem 2 are given two examples when \(S(P)\) is a Gorenstein \(A\)-algebra of finite type, \(S(P)=A[Pt,P^{(2)}t^ 2]\) but with \(P\) a perfect, respectively imperfect, ideal.