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Resolutions of singularities.

Yoshifumi Tsuchimoto

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\fbox{04. definition of singularities}

DEFINITION 04.1   An subvariety $ V=V$ of an affine space $ \mathbb{A}^N$ of codimension $ k$ is non-singular at its point $ P$ if $ V$ is locally defined by $ k$ polynomials $ f_1,\dots, f_k$ such that $ d f_1,\dots d f_k$ is linearly independent at $ P$ .

The dimension and codimensions are defined by using the transcendence degree of the extension of the function field $ k(V)=Q(k[V])$ . The definition of regularity (singularity) is more naturally defined by using theory of commutative algebras.