I’m trying to learn some deformation theory, but I’m stuck on the proof of corollary 4.7 in https://math.berkeley.edu/~robin/math274root.pdf Let $X$ be an affine nonsingular scheme of finite type over some algebraically closed field $k$. Let $D = k[\epsilon]/(\epsilon^2)$, and let $X’$ be an infinitesimal deformation of $X$ over $D$ (ie, the $X’$ is flat over […]

Is it possible to construct a flat family $$ \phi:\mathbb{A}_{\mathbb{C}}^8=\operatorname{Spec} \mathbb{C}[x,y,z,w,a,b,c,d]\longrightarrow \operatorname{Spec} \mathbb{C}[t_1, t_2, t_3] =\mathbb{A}_{\mathbb{C}}^3 $$ so that $$ \phi^{-1}((0,0,0)) = Z(xy+zw,ab+cw+d^2,x+a+c) $$ while $$ \phi^{-1}((t_1, t_2, t_3)) = Z(xy+zw,ab+cw,x+a+c), $$ for some $(t_1, t_2, t_3)\not=(0,0,0)$?

A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y’\supset Y$ where $Y’$ is an infinitesimal thickening of $Y$, then $X$ is non-singular. My question is, if $k$ is algebraically closed, can we say explicitly what every […]

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