In many problems in mathematics, the structure of the set of all orbits of a group acting on a suitable space is described by means of invariants of the group action. In particular, the case of the action of a linear algebraic group G on an algebraic variety X is the case studied in classical invariant theory. The presented book is intended for beginners as a first introduction to the theory. Knowledge of fundamental notions and basic facts from algebraic geometry is expected. The purpose of the book is to offer a short description of main ideas of the classical invariant theory illustrated by many specific examples. Every chapter ends with a set of exercises and with hints for further reading. The first two chapters treat the classical example of the action of the group GL(V) on homogeneous polynomials of degree m on the space E, where E itself is the space of homogeneous polynomials of degree d on V. In the next chapter, the Nagata theorem on the algebra of invariant polynomials on the space of a linear rational representation of a reductive algebraic group is proved. The next chapter is devoted to linear rational representations of a non-reductive algebraic group, including the Nagata counterexample to Hilbert’s 14th problem. Covariants of the action are treated in Chapter 5. Categorical and geometric quotients and linearization of the action, a notion of stability of an algebraic action together with numerical criterion of stability are described next. The last three chapters treat some special cases (hypersurfaces in projective space, ordered sets of linear subspaces in projective space, and toric varieties).

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