On a problem of Griffiths: An inversion of Abel's theorem for families of zero-cycles (Q1426917)

Let \(Y \subset {\mathbb{P}}^3(\mathbb{C})\) be a smooth irreducible surface of degree five, and let \(\Delta_t\) be the line defined by the (affine) equations \(y = a x + b\) and \(z = a' x + b'\), with \(t = (a, a' , b , b')\). For a generic \(t_0 \in \mathbb{C}^4\) and a neighborhood \(U\) of \(t_0\), there are analytic maps \(P_i: U \rightarrow Y\), with \(i = 1, \dots, 5\), such that \(\Delta_t \cap Y = P_{1}(t) + \cdots + P_5(t)\). By Abel's theorem, given a holomorphic two-form \(\omega\) on \(Y\) we get \(\sum_{i = 1}^5 P^*_i(\omega) = 0\). In: Algebraic geometry, The Johns Hopkins centen.\ Lect., Symp.\ Baltimore/Maryland 1976, 26--51 (1977; Zbl 0422.14016), \textit{P. A Griffiths} states a converse to this, asserting that if \(U \subset \mathbb{C}^4\) is an open set, \(U_i\), with \(i = 1, \dots, 5\), are disjoint open sets of \(Y\) and \(P_i: U \rightarrow U_i\), with \(i = 1, \dots, 5\), are holomorphic maps such that \(\sum_{i = 1}^5 P^*_i(\omega) = 0\) for every holomorphic two-form \(\omega\) on \(Y\) then for every \(t \in U\) the zero-cycle \(P_{1}(t) + \cdots + P_5(t)\) can be written as \(\Delta_t \cap Y\), where \(\Delta_t\) is a line in \({\mathbb{P}}^3(\mathbb{C})\). In the paper under review, the author proves a generalized form of Abel's theorem for families of zero-cycles and then proves a result analogous to that in \textit{P. A Griffiths'} paper, here dealing with hypersurfaces in \({\mathbb{P}}^N(\mathbb{C})\) of degree greater than \(N + 1\), where \(N \geq 2\). The paper ends with results specific to the case \(N = 2\), i.e.\ the case where the hypersurface is a curve in \({\mathbb{P}}^2(\mathbb{C})\); among these results there is a classification (previously obtained by Ciliberto [Pubbl. Ist. Mat. ``R.\ Caccioppoli'' Univ.\ Napoli 39, (Naples 1983)]) for linear series of maximal dimension on plane (possibly singular) curves.