by definition, an affine variety is a closed subvariety of some affine space. a classical result asserts that every smooth affine variety of dimension n is isomorphic to a closed subvariety of a $2n+1$-dimensional affine space. given a fixed smooth affine variety x it is natural to ask when x can be realized as a closed subvariety of affine space of dimension $n+d$ for $d < n+1$. in general, there are cohomological obstructions to the existence of such embeddings, and we will discuss such obstructions in the context of homotopy theory of varieties (no prior knowledge of this theory will be assumed).