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
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