Degree bounds for the defining equations of arithmetically Cohen-Macalay varieties
\textit{R. Treger} [Duke Math. J. 48, 35-47 (1981; Zbl 0474.14030)] has shown that the defining prime ideal \(I_ V\) of an arithmetically Cohen- Macaulay non-degenerate variety V of degree s and codimension r may be generated by forms of degree \(\geq ]s/r[\), where \(]s/r[\) denotes the least integer \(\geq s/r\). This paper gives finer degree bounds for the generators of \(I_ V\) which involves the number of independent conditions imposed by V on the hypersurfaces of some fixed degree. As a consequence, we can describe almost all cases where Treger's bound is attained. Namely, suppose that \(s\geq (r+1)^ 2\), then \(I_ V\) needs a generator of degree \(]s/r[\) iff \(V\) lies on a variety of degree r and codimension \(r-1\) (variety of minimal degree), cut out by the quadric hypersurfaces containing V. In particular, if V is a smooth variety with \(s\geq \max \{dr+2,(r+1)^ 2\},\) \(d:=\dim(V)\), then \(I_ V\) needs a generator of degree ]s/r [ iff V is a Castelnuovo variety (i.e. of maximal geometric genus). If V does not lie on any hypersurface of degree t, we can show that \(I_ V\) may be generated by forms of degree \(\leq ]s/r[-[t(t-1)/4]\) (or \(\leq ]s/r[-t(r-1)/2\) if \(r\geq s\) or \(s\geq t^ 3- 3t-1)\).
degree bounds for the defining equations
arithmetically Cohen-Macaulay non-degenerate variety
scientific article; zbMATH DE number 3997994