Tangent Algebras

\noindent We study the Zariski tangent cone $T_X\stackrel{\pi}{\lar} X$ to an affine variety $X$ and the closure $\bar{T}_X$ of $\pi^{-1}({\rm Reg}(X))$ in $T_X$. We focus on the comparison between $T_X$ and $\bar{T}_X$, giving sufficient conditions on $X$ in order that $T_X=\bar{T}_X$. One aspect of the results is to understand when this equality takes place in the presence of the reducedness of the Zariski tangent cone. Our other interest is to consider conditions on $X$ in order that $\bar{T}_X$ be normal or/and Cohen--Macaulay, and to prove that they are met by several classes of affine varieties including complete intersection, Cohen--Macaulay codimension two and Gorenstein codimension three singularities. In addition, when $X$ is the affine cone over a smooth arithmetically normal Calabi--Yau projective variety, we establish when $\bar{T}_X$ is also (the affine cone over) an arithmetically normal Calabi--Yau like (projective) variety.