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When does one use infinitesimal deformation (over the ring of dual numbers $k[t]/(t^2)$) versus local deformation (over $k[t]$ or $k[t_1,\ldots, t_n]$)?

It seems that one works over the ring of dual numbers in order to remove or obtain certain singularities (or to compute for degenerate schemes) while one of the reasons one works over $k[t]$ or $k[t_1,\ldots, t_n]$ is to study and relate fibers over various base points, assuming that the ring of interest is a free $k[t]$ or $k[t_1,\ldots, t_n]$-module.

Would you say that this correct?

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I think what I wrote above is correct since $k[x,y]/(xy)$ has a 1-dimensional space of deformations over the dual numbers.

$$ $$ This link is also helpful.

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