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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].

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