On equations defining arithmetically Cohen-Macaulay schemes. III (Q917641)

[For part II see Duke Math. J. 48, 35-47 (1981; Zbl 0474.14030).] The author characterizes in section 1 the Hilbert function of a Cohen- Macaulay (CM) scheme \(V^ d_ e\subset {\mathbb{P}}^{\nu}\) \((\nu >d\geq 1)\) of given degree e lying on a unique hypersurface of given degree k and such that its general curvilinear section \(V^ 1=V^ d\cap {\mathbb{P}}^{r+1}\), \(r=\nu -d\geq 2\), has minimal (arithmetic) genus (see proposition 1.3). He also describes a more geometric construction of the reduced irreducible codimension two CM varieties \(V^ d_ e\subset {\mathbb{P}}^{\nu}\) whose general curvilinear section has minimal genus among the CM curves of degree e in \({\mathbb{P}}^ 3\) (see the proof of theorem 1.7). In section 2 d-dimensional CM schemes of degree \(e=\left( \begin{matrix} m-1+\nu -d\\ \nu -d\end{matrix} \right)\) in \({\mathbb{P}}^{\nu}\), whose general curvilinear sections have minimal genus, are considered. The corresponding part of the Hilbert scheme is a connected smooth variety which is for \(\nu =d+2\) birationally equivalent to \(\prod^{d+1}_{i=1}Sym^ e({\mathbb{P}}_ i^{\nu -d}) \) in a canonical way (theorem 2.6).