Deformations of pairs \((X,L)\) when \(X\) is singular (Q2845720)

Let \(X\) be a reduced local complete intersection scheme and \(L\) a line bundle on \(X\). The deformation theory of the pair \((X,L)\) was known to the experts; here the author gives an elementary construction. The tangent space is \(\text{Ext}^1_{\mathcal{O}_X} (\mathcal{P}^1_X(L),L)\), where \(\mathcal{P}^1_X(L)\) is the sheaf of principal parts. The obstructions lie in \(\text{Ext}^2\). There is a map \(\text{Ext}^1_{\mathcal{O}_X} (\mathcal{P}^1_X(L),L)\otimes H^0(X,L)\longrightarrow H^1(X,L)\) characterising which sections extend.