Lectures on Curves on an Algebraic Surface

The aim of this book is the study of curves on a smooth projective surface. However, many of the results proved here hold in greater generality. These constitute some of the fundamental theorems in Algebraic Geometry. We find here the construction of the basic geometric objects, such as Grothendieck’s Hilbert scheme, the Picard scheme and the Chow variety. The book also contains the infinitesimal study of the Picard scheme i.e. the determination of the Zariski tangent space, not only for the Picard scheme but also for the subtle case of the reduced Picard scheme (in positive characteristic). The infinitesimal study of a universal curve is also given and this is related to what was called the completeness of the characteristic linear system of a good system of curve.

The first few chapters, give a quick and insightful introduction to the theory of schemes e.g. the definition of schemes, the functorial language, morphisms, sheaves, cohomology of coherent sheaves on the projective space, flat morphisms, some particular cases of the Riemann-Roch theorem.

This book which appeared forty years ago served as an excellent introduction to the ideas and methods of Grothendieck’s new (at that time) theory of schemes. It continues to play the same role even now.