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The contributions making up this volume are expanded versions of the courses given at the C.I.M.E. Summer School on the Theory of Moduli.

This account of deformation theory in classical algebraic geometry over an algebraically closed field presents for the first time some results previously scattered in the literature, with proofs that are relatively little known, yet relevant to algebraic geometers. Many examples are provided. Most of the algebraic results needed are proved. The style of exposition is kept at a level amenable to graduate students with an average background in algebraic geometry.

In one sense, deformation theory is as old as algebraic geometry itself: this is because all algebro-geometric objects can be deformed by suitably varying the coef?cients of their de?ning equations, and this has of course always been known by the classical geometers. Nevertheless, a correct understanding of what deforming means leads into the technically most dif?cult parts of our discipline. It is fair to say that such technical obstacles have had a vast impact on the crisis of the classical language and on the development of the modern one, based on the theory of schemes and on...

An undergraduate textbook suitable for linear algebra courses. This text develops the linear algebra hand in hand with the geometry of linear (or affine) spaces in such a way that the understanding of each reinforces the other.