La science et l'hypothèse
《La science et l'hypothèse》 · 第 286–287 页 · Section 8.1，Definition 8.1.4(a) 及共同对象条件
Let \(X\) be a compact, connected, oriented, smooth 4-manifold. For \(\partial X=\varnothing\), a Lefschetz pencil on \(X\) is a nonempty finite subset \(B\) of \(X\), called the base locus, together with a smooth map \(\pi:X-B\to\mathbb{CP}^1\) such that each point \(b\in B\) has an orientation-preserving coordinate chart in which \(\pi\) is given by projectivization \(\mathbb C^2-\{0\}\to\mathbb{CP}^1\), and each critical point of \(\pi\) has an orientation-preserving chart on which \(\pi(z_1,z_2)=z_1^2+z_2^2\) relative to a suitable smooth chart on \(\mathbb{CP}^1\). For \(t\in\mathbb{CP}^1\), the fiber \(F_t\) is \(\pi^{-1}(t)\cup B\subset X\).