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Suppose X, Y, Z are k-varieties and $f: X \to Z$ factors through $f': X \to Y$ and $g: Y \to Z$. Suppose all of f, f', g are surjective. Assume that for $z \in Z$, the fibre $f^{-1} (z)$ is reduced. Then, is the fibre $g^{-1} (z)$ always reduced? If not, when will it be true? |
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6
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How about: $Y$ is the cubic curve $y^2=x^2+x^3$ in $\mathbb A^2$ minus the point $(-1,0)$, $g:Y\to Z$ is the projection to the $x$-axis, and $X$ is the normalization of $Y$, which would be $\mathbb P^1$ minus 2 points. The fiber $g^{-1}(0)$ is one non-reduced point. The fiber $f^{-1}(0)$ is two reduced points. |
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5
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The fiber $X_z$ is equal to $X\times_Y Y_z$. So if $X\to Y$ is faithfully flat, then $X_z$ reduced implies that $Y_z$ is reduced. |
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