The Tangent Space and Smoothness
In this chapter we introduce the tangent space at a point and review several criteria for smoothness for arbitrary varieties. It is well known that the tangent space at a point of an irreducible variety always has dimension at least as large as the dimension of the variety. Furthermore, a variety is smooth at a point if the dimensions are equal. If one knows equations defining the variety, then the Jacobian criterion can be used to determine the dimension of the tangent space at a point. In this chapter we also recall the notion of multiplicity at a given point on an algebraic variety and relate this to another criterion for smoothness. There are still many other ways for determining smoothness in Schubert varieties which will be discussed in the remaining chapters of this book. A summary of all criteria for smoothness and rational smoothness appears on Page 208.
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