Generators and relations for finitely generated graded normal rings of dimension two (Q1111627)

After Pinkham, a finitely generated graded normal ring \(R=\oplus R_ k \) of dimension 2 over \({\mathbb{C}}\) \((R_ 0={\mathbb{C}})\) is isomorphic to \({\mathcal L}(D)=\oplus^{\infty}_{n=0}L(nD), \) where D is a ``fractional'' divisor on a Riemann surface X of genus g, i.e. \(D=D_ 0+\sum^{k}_{i=1}\gamma_ iP_ i, \) \(D_ 0=\sum_{P\in X}n_ PP, \) \(n_ P\in {\mathbb{Z}}\), all but finitely many \(n_ P=0\), \(P_ i\in X\), \(\gamma_ i\in {\mathbb{Q}}\), \(0<\gamma_ i<1\) and L(nD) denotes the set of meromorphic functions f such that \(div(f)+nD\geq 0\). The present paper shows that in many cases a set of homogeneous generators of \({\mathcal L}(D)\) and their relations can be determined if a set of homogeneous generators and their relations are known for \({\mathcal L}(D_ 1)\), where \(D_ 1<D\) and \({\mathcal L}(D_ 1)\) has a much simpler structure than \({\mathcal L}(D)\). In particular the degrees of homogeneous generators and relations can be found for all D for which \(\deg (D_ 0)\geq 2g+1\) or \(D_ 0\) is the canonical divisor on a nonhyperelliptic curve of genus \(g>3\) (this extends some results of Mumford and Saint-Donat).