Line bundles and syzygies of trigonal curves
Let \({\mathbb{C}}\) be a trigonal curve of genus \(p>4\) over the field of complex numbers, i.e. a nonhyperelliptic curve admitting a 3-sheeted covering map \(\pi:{\mathbb{C}}\to {\mathbb{P}}_ 1\) with ramification divisor of degree \(2p+4.\) Let T denote the unique trigonal line bundle andm the Maroni invariant of \({\mathbb{C}}\). m can be defined as follows: If X is the rational scroll associated to the canonical model of \({\mathbb{C}}\) in \({\mathbb{P}}^{p-1}\), then m is the degree of the minimal directrix of X [\textit{A. Maroni}, Ann. Mat. Pura Appl., IV. Ser. 25, 341-354 (1946; Zbl 0061.354)]. The main result of the paper is the following statement about the syzygies of a very ample nonspecial line bundle of degree 2p-k with \(0\leq k\leq m\) on \({\mathbb{C}}:\) for an integer \(i\) with \(0\leq i\leq m-k\) the graded Betti number \(\beta_{i,i+2}\) associated to L vanishes if and only if \(H^ 0(L\otimes K_{{\mathbb{C}}}^{-1}\otimes T^{k+i})=0\). Furthermore \(\beta_{i,i+2}\neq 0\) for \(m-k+1\leq i\leq r+1\). Note that according to a theorem of \textit{M. L. Green} [J. Differ. Geom. 19, 125-171 (1984; Zbl 0559.14008)] the Betti numbers \(\beta_{ij}\) vanish apart from \(\beta_{\infty}(=1)\), \(\beta_{i,i+1}\) (which is \(\neq 0\) for \(i=1,...,h^ 0(L)-2)\), and the above \(\beta_{i,i+2}\). As special cases of the main result one gets that under the above hypotheses L is not normally generated if and only if \(L=K_{{\mathbb{C}}}\otimes T^{-k}(P_ 1+...+P_{2k+2})\) for some points \(P_ 1,...,P_{2k+2}\) on \({\mathbb{C}}\) (Improving a theorem of \textit{M. L. Green} and \textit{R. Lazarfeld} [Invent. Math. 83, 73-90 (1986; Zbl 0594.14010)] in the case of a trigonal curve) and for \(k<m\), L is not normally presented, if and only if \(L=K_{{\mathbb{C}}}\otimes T^{-(k+1)}(Q_ 1+...+Q_{2k+5})\) for some points \(Q_ 1,...,Q_{2k+5}\) on \({\mathbb{C}}\). The proof uses the scroll corresponding to the trigonal covering associated to the line bundle L and relies on the method of [\textit{F.-O. Schreyer}, Math. Ann. 275, 105- 137 (1986; Zbl 0578.14002)].
scientific article; zbMATH DE number 4021336