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Chapters on algebraic surfaces. (English) Zbl 0910.14016

Kollár, János (ed.), Complex algebraic geometry. Lectures of a summer program, Park City, UT, 1993. Providence, RI: American Mathematical Society. IAS/Park City Math. Ser. 3, 5-159 (1997).
These notes represent a graduate course given by the author in several circumstances at the Universities of Warwick, East China Normal University, Shanghai, University of Utah, as well as at the Park City summer school. They aim to introduce the reader to some fascinating topics from the classification of surfaces, at a research level. The notes contain four chapters (chapters 1-4) and five further chapters (chapters A-E). Some of the chapters have a number of well chosen exercises. In the first two chapters some examples of surfaces are discussed: the cubic surface (lines, the lattice and the intersection pairing, the divisor class group, the conic bundle structure, the cubic surface as \(\mathbb{P}^2\) blown up six times, etc.) in chapter 1, and the rational scrolls (from a classical point of view) in chapter 2.
In chapter A the author introduces the intersection numbers of divisors, gives various examples and briefly discusses how to prove Bezout’s theorem. Then one shows that the intersection matrix of a connected curve that blows down to a point is negative definite, as well as the fact that the intersection matrix of the components of a fibre of a morphism from a surface to a curve is negative semidefinite. Finally, the canonical class is introduced and the adjunction formula is proved. – In chapter B one introduces the concept of sheaf in algebraic geometry and one discusses the axiomatic definition of coherent cohomology, Serre’s vanishing theorem and the Serre duality.
Chapter 3 is devoted to K3 surfaces. One of the author’s reasons to do that before the classification of surfaces is the following “K3s occupy a special place in the curriculum for anyone trying to master algebraic surfaces... They are marvellous testing ground for understanding of linear systems, cohomology, vanishing theorems, the structure sheaf \(\mathcal O_D\) of a nonreduced divisor \(D\), the relation between geometry of linear systems and the algebra of graded rings, singularities, intersection numbers of curves and quadratic forms, Hodge structures, moduli, and many other things.” One also gives the classical definition of the K3s, as those surfaces in \(\mathbb{P}^g\) having a hyperplane section that is a canonical curve \(C\subset \mathbb{P}^{g-1}\).
Chapter 4 deals with isolated singularities of surfaces. Here a thorough study of rational singularities, Gorenstein two-dimensional singularities, with special emphasis to elliptic Gorenstein surface singularities (also called minimally elliptic singularities in the literature), and Artin’s contractibility criterion. Many examples of singularities are given.
Chapter D deals with the minimal model problem for surfaces, presented in an elegant modern way, from the point of view of Mori’s theory. – The last chapter (chapter E) presents the strategy of the classification of surfaces whose canonical class is nef. Many of the original ideas of Enriques are explained in a marvellous way: The author insists especially in underlining the main ideas that are beyond the classification. Although not completely selfcontained, all in all, these notes are extremely useful for anyone wanting to introduce himself to the classification of surfaces.
For the entire collection see [Zbl 0866.00043].

MSC:

14J10 Families, moduli, classification: algebraic theory
14E30 Minimal model program (Mori theory, extremal rays)
14J25 Special surfaces
14J28 \(K3\) surfaces and Enriques surfaces
14J17 Singularities of surfaces or higher-dimensional varieties
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