On the graded Betti numbers of plane algebraic curves (Q1812655)

The author considers an assumed connection between the Clifford index \(c\) and the graded Betti numbers \(\beta_{ik}\) of a Riemann surface (algebraic curve) \(C\), namely Green's conjecture: \(c=\min\{i:\beta_{ii+1}\neq 0\}\). Theorem 1. Let \(C\subset\mathbb{P}^ 2\) be a smooth, plane algebraic curve. Then Green's conjecture is true for \(C\). Theorem 2. Let \(C\subset\mathbb{P}^ 3\) be a smooth complete intersection of type \((a,b)\) with \(a+b\leq 6\). Then Green's conjecture holds.