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Let $S$ be a complex non-singular projective surface and let $C$ be a complex non-singular projective curve. Moreover consider a morphism $\varphi:S\longrightarrow C$ which is flat, proper and with connected fibers.

Is it true that the subset $$U=\{p\in C\,:\, \text{the fiber $S_p$ is a smooth curve}\}\subseteq C$$ is an open subset of $C$?

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  • $\begingroup$ As I explained in an answer to your earlier question, the hypotheses "flat" and "proper" are redundant here. It is important to understand what technical conditions mean, and whether they are actually doing anything in a given context. Anyway, for a more down-to-earth reference than the one given by @IrfanKadikoylu, see Theorem II.6.2 in Shafarevich. $\endgroup$ – user64687 Jan 16 '15 at 8:42
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Yes that is true. See Hartshorne Ch.III Corollary 10.7. In fact, you don't need half of the assumptions above, it is a more general fact. It works if $S$ is nonsingular surface over $\mathbb{C}$ and $C$ is any complex curve and $\varphi$ is a separated morphism with connected fibers.

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