(Globally generated)
\(\mcf\) is generated by global sections if there are \(\ts{s_i}_{i \in I} \subseteq \Gamma(X, \mcf)\) whose images in \(\mcf_x\) generate that stalk as an \(\OO_{X,x}\)-module for every \(x\). Equivalently, there is a surjection \(\bigoplus_{i \in I} \OO_X \surjects \mcf\).
Every quasicoherent sheaf on an affine scheme is globally generated by any generating set of its corresponding module. For \(n\ge1\), \(\OO_{\PP^n}(d)\) is globally generated if and only if \(d \geq 0\): for \(d\ge0\) the monomials of degree \(d\) have no common zero, and for \(d < 0\) there are no nonzero global sections.
Let \(X\) be a scheme over a ring \(A\) and \(\mcl\) an invertible sheaf generated by global sections \(s_0,\ldots,s_n\). There is a unique \(A\)-morphism \(f\colon X \to \PP^n_A\) with \(f^*\OO(1)\cong\mcl\) and \(f^*x_i=s_i\) [@Har10a, Theorem II.7.1]; \(\mcl\) is very ample over \(\Spec A\) if and only if some such \(f\) is an immersion.
Dependencies
D-QNTZYQuasicoherent and coherent sheavesuses
Backlinks
D-MODAMPLEAmple and very ample invertible sheavesusesT-MODSERRESerre's theorems A and Buses