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Let $X$ be a non-reduced scheme. Denote by $X_\mathrm{red}$ the reduced scheme associated to $X$. Assume further that $X_\mathrm{red}$ is smooth. Let $Z$ be a reduced scheme and $f:X_\mathrm{red} \to Z$ be a proper morphism. Does there exist a push-out of $X$ and $Z$ along $X_\mathrm{red}$? In other words does there exist a scheme $W$ along with morphisms from $X, Z$ to $W$ such that $$\begin{array}{lcr} X_\mathrm{red}&\to&X\\ \downarrow & \square &\downarrow\\ Z &\to &W\end{array}$$is a Cartesian diagram and $W$ satisfies the universal property of push-out?

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  • $\begingroup$ "Assume further that $X_{red}$ is smooth": are you working a field? $\endgroup$ Aug 25, 2013 at 19:29
  • $\begingroup$ @Romagny: Over the complex numbers $\endgroup$
    – Chen
    Aug 25, 2013 at 19:34
  • $\begingroup$ I still don't really have ideas (although I am rather pessimistic), but I now wonder: why do you require the diagram to be cartesian? Usually cocartesian diagrams are not cartesian, and there is no reason to require that a pushout diagram be. $\endgroup$ Apr 19, 2014 at 19:39

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