Let $k$ be a field. Given a morphism of schemes $\operatorname{Spec}k[T,U,V,W]/((U+T)W,(U+T)(U^3+U^2+UV^2-V^2))\to \operatorname{Spec}k[T]$. When is the fibre is reduced or irreducible?
I tried to solve it as follows.
The corresponding ring of the fibre at a point $y\in \operatorname{Spec}k[T]$ is $$k[T,U,V,W]/((U+T)W,(U+T)(U^3+U^2+UV^2-V^2))\otimes_{k[T]}\kappa(y)=\kappa(y)[U,V,W]/((U+\bar{T})W,(U+\bar{T})(U^3+U^2+UV^2-V^2).$$ $\kappa(y) $ is the residue field. But I don't know how to continue. Could anyone help me? Thank you.