Introduction

This article 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,

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.

Part I is primarily expository. It is intended to give a fairly

complete picture of problems arising from the Noether-Lefschetz

theorem and of the state of our knowledge at the present time.

In particular, in § 1 we discuss the most popular approaches to

proving the Noether-Lefschetz theorem.

In § 2 we give an exposition of a series of natural questions

about the locus of smooth surfaces in

P3

of degree d with Picard

group not generated by 0(1) (the so called Noether-Lefschetz

locus). Many of these questions have been answered only recently

vn