Book
Computations in algebraic geometry with Macaulay 2
Here is our book, Computations in algebraic geometry with Macaulay 2, edited by David Eisenbud, Daniel R. Grayson, Michael E. Stillman, and Bernd Sturmfels. It was published by SpringerVerlag in September 25, 2001, as number 8 in the series "Algorithms and Computations in Mathematics", ISBN 3540422307, price DM 79,90 (net), or $44.95.
 Errata
 The Macaulay2 code used in the book is available in machine readable form using subversion at svn://macaulay2.math.uiuc.edu/Macaulay2/trunk/M2/Macaulay2/packages/ComputationsBook, with one subdirectory per chapter. Each file */chapter.m2 contains the original code from the chapter, whereas the file */test.m2 contains code slightly updated to run properly with the latest version of Macaulay2. The chapter code files are also available as part of the tar file containing the source code.
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 The entire book, except for the cover and the front matter:
Available in the following formats: dvi (with needed figures: octahedron.eps, standardpairsfixed.eps, and Y25.eps), postscript, pdf.  The cover and the front matter.
 Preface:
Available in the following formats: dvi, postscript, pdf.  Part I, Introducing Macaulay 2

Ideals, Varieties, and Macaulay 2, by Bernd Sturmfels:
Available in the following formats: dvi, postscript, pdf.
This chapter introduces Macaulay2 commands for some elementary computations in algebraic geometry. Familiarity with Gröbner bases is assumed. 
Projective Geometry and Homological Algebra, by David Eisenbud:
Available in the following formats: dvi, postscript, pdf.
We provide an introduction to many of the homological commands in Macaulay2 (modules, free resolutions, Ext and Tor, ...) by means of examples showing how to use homological tools to study projective varieties. 
Data Types, Functions, and Programming, by Daniel Grayson and Michael Stillman:
Available in the following formats: dvi, postscript, pdf.
In this chapter we present an introduction to the structure of Macaulay2 commands and the writing of functions in the Macaulay2 language. For further details see the Macaulay2 manual distributed with the program. 
Teaching the Geometry of Schemes, by Gregory Smith and Bernd Sturmfels:
Available in the following formats: dvi, postscript, pdf.
This chapter presents a collection of graduate level problems in algebraic geometry illustrating the power of Macaulay2 as an educational tool.

Ideals, Varieties, and Macaulay 2, by Bernd Sturmfels:
 Part II, Mathematical Computations

Monomial Ideals, by Serkan Hosten and Gregory Smith:
Available in the following formats: dvi (with needed figures: octahedron.eps, and standardpairsfixed.eps), postscript, pdf.
Monomial ideals form an important link between commutative algebra and combinatorics. In this chapter, we demonstrate how to implement algorithms in Macaulay2 for studying and using monomial ideals. We illustrate these methods with examples from combinatorics, integer programming, and algebraic geometry. 
From Enumerative Geometry to Solving Systems of Polynomial
Equations, by Frank Sottile:
Available in the following formats: dvi (with needed figure: Y25.eps), postscript, pdf.
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerative Geometry is concerned with counting the number of solutions when the polynomials come from a geometric situation and Intersection Theory\index{intersection theory} gives methods to accomplish the enumeration.
We use Macaulay2 to investigate some problems from enumerative geometry, illustrating some applications of symbolic computation to this important problem of solving systems of polynomial equations. Besides enumerating solutions to the resulting polynomial systems, which include overdetermined, deficient, and improper systems, we address the important question of real solutions to these geometric problems. 
Resolutions and Cohomology over Complete Intersections,
by Luchezar Avramov and Daniel Grayson:
Available in the following formats: dvi, postscript, pdf.
This chapter contains a new proof and new applications of a theorem of Shamash and Eisenbud, providing a construction of projective resolutions of modules over a complete intersection. The duals of these infinite projective resolutions are finitely generated differential graded modules over a graded polynomial ring, so they can be represented in the computer, and can be used to compute Ext modules simultaneously in all homological degrees. It is shown how to write Macaulay2 code to implement the construction, and how to use the computer to determine invariants of modules over complete intersections that are difficult to obtain otherwise. 
Algorithms for the Toric Hilbert Scheme,
by Michael Stillman, Bernd Sturmfels, and Rekha Thomas:
Available in the following formats: dvi, postscript, pdf.
The toric Hilbert scheme parametrizes all algebras isomorphic to a given semigroup algebra as a multigraded vector space. All components of the scheme are toric varieties, and among them, there is a fairly well understood coherent component. It is unknown whether toric Hilbert schemes are always connected. In this chapter we illustrate the use of Macaulay2 for exploring the structure of toric Hilbert schemes. In the process we will encounter algorithms from commutative algebra, algebraic geometry, polyhedral theory and geometric combinatorics. 
Sheaf Algorithms Using the Exterior Algebra,
by Wolfram Decker and David Eisenbud
Available in the following formats: dvi, postscript, pdf.
In this chapter we explain constructive methods for computing the cohomology of a sheaf on a projective variety. We also give a construction for the Beilinson monad, a tool for studying the sheaf from partial knowledge of its cohomology. Finally, we give some examples illustrating the use of the Beilinson monad. 
Needles in a Haystack: Special Varieties via Small Fields,
by Frank Schreyer and Fabio Tonoli:
Available in the following formats: dvi, postscript, pdf.
In this article we illustrate how picking points over a finite field at random can help to investigate algebraic geometry questions. In the first part we develop a program that produces random curves of genus g < 15. In the second part we use the program to test Green's Conjecture on syzygies of canonical curves and compare it with the corresponding statement for Coble selfdual sets of points. In the third section we apply our techniques to produce CalabiYau 3folds of degree 17 in P^{6}. 
Dmodules and Cohomology of Varieties, by Uli Walther:
Available in the following formats: dvi, postscript, pdf.
In this chapter we introduce the reader to some ideas from the world of differential operators. We show how to use these concepts in conjunction with Macaulay2 to obtain new information about polynomials and their algebraic varieties.

Monomial Ideals, by Serkan Hosten and Gregory Smith: