Moishezon's theorem and degeneration (Q2333387)

Lefschetz proved M. Noether's statement: For very general surfaces \(S \subset {\mathbb{ P}}^3_{\mathbb{C}}\) of degree \( d \geq 4\), the restriction map \( r: {\mbox{Pic}} {\mathbb{ P}}^3_{\mathbb{C}} \rightarrow {\mbox{Pic}} S \) is an isomorphism. If dim\( X > 3\), \(X\) is smooth and \(Y \subset X\) is any effective ample divisor, then \(r : {\mbox{Pic}} X \rightarrow {\mbox{Pic}} Y\) is an isomorphism by Grothendieck-Lefschetz theorem. When dim \(X = 3\), the problem is more subtle and many results have been received if \(K \otimes L\) is generated by its global sections or the multiplication map \( H^0(K \otimes L) \otimes H^0(L) \rightarrow H^0(K \otimes L \otimes L)\) is surjective, where \(L\) is a line bundle on \(X\). In this paper, the authors prove Theorem 1. Let X be a smooth projective complex threefold. If \(A, B \in {\mbox{Pic}} X\) are very ample and \( H^0(K_X(A \otimes B)) \neq 0\), then \( r: {\mbox{Pic}} X \rightarrow {\mbox{Pic}} Y\) is an isomorphism for very general \( Y \in |A \otimes B|\). Theorem 1 not only recovers the theorem of Moishezon-Voisin for line bundles L that are decomposable as a tensor product of very ample line bundles, but also has the following important application. Corollary 2. Let \( X \subset {\mathbb{ P}}^n_{\mathbb{C}}\) be a smooth projective threefold. If \(d > 1\) and \( H^0(K_X(d)) \neq 0\), then the restriction \( r : {\mbox{Pic}} X \rightarrow {\mbox{Pic}} Y\) is an isomorphism for very general \(Y \in |O_X(d)|\). To prove Theorem 1, the authors use the method for \( X = {\mathbb{ P}}^3\) developed by Griffiths and Harris, who carefully studied the degeneration of a degree \(d\) surface \(Y\) to a reducible surface \( T \cup P\) in which deg\( T = d - 1 \) and \( P\) is a plane.