This book deals with the study of curves lying on general members of families of smooth projective surfaces over the complex numbers. The guiding philosophy is that the set of curves on such surfaces is as small as it can possibly be; more precisely, this means that the group of classes of Cartier divisors (or, equivalently the group of line bundles called the Picard group) of a general surface has the lowest possible rank given by the geometry of the family. The focus of the book is Noether-Lefschetz theory, the study of the locus of smooth surfaces in P3 whose Picard group is not Z. The first part of the book presents a brief survey of basic concepts and results, together with some natural questions arising in the theory. In the second part, a deformation-theoretic technique introduced by Griffiths and Harris is used to determine the Picard group of a general surface in Z3 containing a fixed curve. This idea is generalized in the third part to families of surfaces in higher projective spaces, namely complete intersection surfaces in Pn and projectively Cohen-Macauley surfaces in P4.
Table of contents:
Part I: A brief survey of Noether-Lefschetz problems; The Neother-Lefschetz theorem; Components of the Neother-Lefschetz locus; Higher dimensional problems; Part II: The Picard group of a general surface containing a space curve; Some classical facts; The results; Part III: The Picard group of some general surfaces in Pr; The generalization of Theorem II.3.1 in Pn+2; The Picard group of general residual surfaces in P4; The Picard group of general projectively Cohen-Macaulay surfaces in P4.
Table of contents:
Part I: A brief survey of Noether-Lefschetz problems; The Neother-Lefschetz theorem; Components of the Neother-Lefschetz locus; Higher dimensional problems; Part II: The Picard group of a general surface containing a space curve; Some classical facts; The results; Part III: The Picard group of some general surfaces in Pr; The generalization of Theorem II.3.1 in Pn+2; The Picard group of general residual surfaces in P4; The Picard group of general projectively Cohen-Macaulay surfaces in P4.