On the equations defining quasi complete intersection space curves (Q1383696)

A non-complete intersection smooth irreducible subvariety \(X\subset\mathbb{P}^n\), \(n\geq 3\), of codimension 2 is called a ``quasi complete intersection'' (qci for short), if it is scheme-theoretic intersection of 3 hypersursurfaces \(F_1,F_2,F_3\); i.e. there is a surjective morphism \[ \bigoplus^3_{i=1} {\mathcal O}(-a_i) @>\Phi=(F_1,F_2,F_3)>> {\mathcal I}_X\to 0 \] where \(a_1\leq a_2\leq a_3\) are the degrees of the hypersuperfaces \(F_i\)'s. Such varieties have been extensively studied by \textit{M. Fiorentini} and \textit{A. T. Lascu} [see Rend. Sem. Mat. Fis. Milano 57, 161-182 (1987; Zbl 0704.14040) for a survey on results and related bibliography]. The first result of the paper under review is that the degrees \(a_1,a_2,a_3\) are determined by \(X\), namely if \(X\) is qci of three hypersurfaces of degrees \(b_1\leq b_2\leq b_3\), then \((a_1,a_2,a_3) =(b_1,b_2,b_3)\) (see proposition 1). If \(n=3\), this result is obtained by the cohomology of the rank 2 vector bundle \(\text{Ker} (\Phi)\), via a result of \textit{P. Rao} [J. Algebra 86, 23-34 (1984; Zbl 0528.14008)], while the general case is reduced to the previous one by means of the intersection of \(X\) with a general projective 3-space. Another problem regarding \(X\) is the relationship between the homogeneous ideal \(\langle F_1,F_2,F_3 \rangle\) and the saturated homogeneous ideal \(I(C)= \bigoplus_{t\in \mathbb{Z}}H^0 (I(t))\). This is studied in case of a smooth connected curve \(C\) in \(\mathbb{P}^3\): If \(C\) is qci of 3 surfaces \(F_1,F_2,F_3\), then the polynomials \(F_i\)'s belong to a minimal system of generators of \({\mathcal I}(C)\) (see proposition 3). The proof uses the fact that the complete intersection \(F_i\cap F_j\) links \(C\) to a subcanonical curve and then the result follows from the analysis of the exact sequences of liaison. This improves a result of \textit{D. Portelli} and \textit{W. Spangher} [J. Pure Appl. Algebra 98, No. 1, 83-93 (1995; Zbl 0833.14001)]. Next a new proof of a formula of enumerative geometry is given (proposition 5): \[ \deg(C) (a_1+a_2+ a_3)-2\text{g} (C)+2=a_1a_2a_3, \] where \(C\) is a smooth connected qci space curve. This relation was obtained by \textit{M. Fiorentini} and \textit{A. T. Lascu} [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 27, 201-227 (1981; Zbl 0513.14036)]. Finally, as a consequence of the previous results it is shown that the general elliptic space curve of degree \(d\geq 15\) is not a qci (see proposition 6) and, more generally, that for any \(g\geq 0\), there is a suitable integer \(d(g)\geq g+3\), such that the general smooth connected space curve of degree \(d\) and genus \(g\) is not a gci (see proposition 7).