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Syzygies of points in projective space and applications. (English) Zbl 0840.14032
Orecchia, Ferruccio (ed.) et al., Zero-dimensional schemes. Proceedings of the international conference held in Ravello, Italy, June 8-13, 1992. Berlin: de Gruyter. 145-170 (1994).
Let \(X\) be a smooth complex projective curve of genus \(g\) and \(L\) a base-point-free line bundle on \(X\), and put \(H^0 (X,L) =V\) and \(h^0 (X,L)= r+1\). For the graded module \(B= \bigoplus_{q\in \mathbb{Z}} B_q= \bigoplus_{q\in \mathbb{Z}} H^0 (X, L^{\otimes q})\) over the graded ring \(S(V)= \bigoplus_{p\geq 0} S_p(V) \cong \mathbb{C}[T_0, \dots, T_r]\), the Koszul cohomology module \(K_{p,q} (X,L)= K_{p,q} (B,V)\) is defined by \(K_{p,q} (B,V)= \text{Ker} (d_{p,q})/ \text{Im} (d_{p+1, q-1})\), where \(d_{p,q}: \bigwedge^p V\otimes B_q\to \bigwedge^{p-1} V\otimes B_{q+1}\) is the map in the Koszul complex. Then the \((M_q)\)-conjecture of M. Green and R. Lazarsfeld states that if \(\deg (L)\gg 2g\), then the property \(M_q\) (which means \(K_{j,1} (X,L)=0\) for \(j\geq r-q\)) fails if and only if there exists a one-dimensional linear series \(g^1_q\) of degree \(q\) on \(X\). The author introduces certain “syzygy” varieties \(V_{\text{Syz}} {}^p(X)\) whose definition is too technical to be stated here, and the main theorem of this paper (which implies \((M_q)\)-conjectures for \(g\geq 13\) and \(q=3\)) is stated as follows:
Theorem. Assume that \(g\geq 13\), \(\deg (L)\geq 2g+3\), \(h^0 (X,L)= r+1\), and \(K_{r-3,1} (X,L)\neq 0\). Then \(V_{\text{Syz}} {}^{r-3}(X)\) is a 3-dimensional rational normal scroll of degree \(r-2\) and the one-dimensional linear system \(\{\mathbb{P}^2_t\mid t\in \mathbb{P}^1\}\) of planes on \(V_{\text{Syz}} {}^{r-2} (X)\) cuts out a \(g^1_3\) on \(X\).
For the entire collection see [Zbl 0797.00007].

14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
13D02 Syzygies, resolutions, complexes and commutative rings
14J30 \(3\)-folds