Preface

The systematic use of Koszul cohomology computations in algebraic geometry

can be traced back to the foundational paper [Gre84a] by M. Green. In this

paper, Green introduced the Koszul cohomology groups Kp,q(X, L) associated to a

line bundle L on a smooth projective variety X, and studied the basic properties

of these groups. Green noted that a number of classical results concerning the

generators and relations of the (saturated) ideal of a projective variety can be

rephrased naturally in terms of vanishing theorems for Koszul cohomology, and

extended these results using his newly developed techniques. In a remarkable series

of papers, Green and Lazarsfeld further pursued this approach. Much of their work

in the late 80’s centers around the shape of the minimal resolution of the ideal of

a projective variety; see [Gre89], [La89] for an overview of the results obtained

during this period.

Green and Lazarsfeld also stated two conjectures that relate the Koszul coho-

mology of algebraic curves to two numerical invariants of the curve, the Clifford

index and the gonality. These conjectures became an important guideline for future

research. They were solved in a number of special cases, but the solution of the

general problem remained elusive. C. Voisin achieved a major breakthrough by

proving the Green conjecture for general curves in [V02] and [V05]. This result

soon led to a proof of the conjecture of Green–Lazarsfeld for general curves [AV03],

[Ap04].

Since the appearance of Green’s paper there has been a growing interaction

between Koszul cohomology and algebraic geometry. Green and Voisin applied

Koszul cohomology to a number of Hodge–theoretic problems, with remarkable

success. This work culminated in Nori’s proof of his connectivity theorem [No93].

In recent years, Koszul cohomology has been linked to the geometry of Hilbert

schemes (via the geometric description of Koszul cohomology used by Voisin in her

work on the Green conjecture) and moduli spaces of curves.

Since there already exists an excellent introduction to the subject [Ei06], this

book is devoted to more advanced results. Our main goal was to cover the re-

cent developments in the subject (Voisin’s proof of the generic Green conjecture,

and subsequent refinements) and to discuss the geometric aspects of the theory,

including a number of concrete applications of Koszul cohomology to problems in

algebraic geometry. The relationship between Koszul cohomology and minimal res-

olutions will not be treated at length, although it is important for historical reasons

and provides a way to compute Koszul cohomology by computer calculations.

Outline of contents. The first two chapters contain a review of a number of

basic definitions and results, which are mainly included to fix the notation and

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