Classification of toric curves in projective space, Rao module and liaison
A toric curve \(C\) in \({\mathbb P}^3\) is given by a parameterization \(x = s^a\), \(y = s^{a-c}t^c\), \(z = s^{a-b}t^b\), \(w = t^a\) with \(a,b,c \in {\mathbb N}\) relatively prime positive integers. If \(a>b>c>0\) then \(C\) is called a \textit{monomial curve}. The paper addresses the following question. Given a toric curve \(C\) in \({\mathbb P}^3\), is there another toric curve \(C'\) linked to \(C\) in an even (resp. odd) number of steps? For the case of monomial curves and linking in an even number of steps (i.e.\ the case of even liaison classes of monomial curves), this was answered by \textit{H. Bresinsky} and \textit{C. Huneke} [J. Reine Angew. Math. 365, 33-66 (1986; Zbl 0574.14029)]. Using an explicit minimal generating set of the ideal \(I(C)\) due to the second author, the authors find the minimal free resolution of \(I(C)\) (including the maps). This allows them to compute the Hartshorne-Rao module \(M(C)\) and conclude about the liaison class. In particular they prove a conjecture of Bresinsky and Huneke [op. cit.] that if \(C\) is a monomial curve such the number of minimal generators of \(I(C)\) is at least 5 then there is no monomial curve linked to \(C\) in an odd number of steps.
scientific article; zbMATH DE number 1271336
scientific article