A remark on the syzygies of the generic canonical curves
If C is a genus g nonhyperelliptic curve, then it is defined the canonical ring R as \(\oplus_{n}H^ 0(\omega^ n_ C)\quad which\) is a graded module over the polynomial ring \(S=Symm(H^ 0(\omega_ C))\). Then it is defined a minimal graded Koszul resolution of \(S/(irrelevant\quad ideal)\) and therefore the Koszul cohomology group \(K_{pq}(R)=K_{pq}(C)\). The paper, extending results of \textit{M. Green} and \textit{R. Lazarsfeld} [J. Differ. Geom. 19, 168-171 (1984; Zbl 0559.14008)] and using results of Bayer and Stillman based on the computer program Macaulay \(proves:\) K\({}_{p2}(X)=0\), when X is a curve of genus n, implies that \(K_{p2}(C)=0\) for a generic curve of genus \(m,\) provided that \(m=n mod(p+1)\) and \(m>n-1;\) K\({}_{22}(C)=0\) for \(g>6\) and \(K_{32}(C)=0\) for \(g>8\).
scientific article; zbMATH DE number 4027657
scientific article