This question deals with a concrete exercise from [Geomerty of Schemes](https://www.google.de/url?sa=t&source=web&rct=j&opi=89978449&url=https://www.maths.ed.ac.uk/%7Ev1ranick/papers/eisenbudharris.pdf&ved=2ahUKEwjzsZi6rf-AAxVKxQIHHah3B6oQFnoECBUQAQ&usg=AOvVaw3usdrlkTi1DgXnq_xqASCS) by Eisenbud and Harris but also moreover the general philosophy attacking typical problems in algebraic geometry of following relative nature: say we have a "nice" enough map (having here sloppy said something "fibration like" in mind) $f:X \to Y$ of schemes over base field $K$, and assume $Y$ (wlog we can assume it to be affine) has some property $\mathcal{P}$ and say we have additional assumpion that there exist a fiber $X\_y$ which has also property $\mathcal{P}$.


Natural question: in which situations one should "expect" that this property is carried/ "inherited" by neighbored fibers. Or even stronger, the property $\mathcal{P}$ inherited by $X$ *generically*? Let's consider the concrete problem from the book:


**Exercise III-74.** Let $K$ be a field, and let $ B =\mathbb{A}^{12}\_K =\operatorname{Spec}K[a,b,c,d,e,f,g,h,i,j,k,l]$. Consider the two conic curves $\mathcal{C}\_i \subset \mathbb{P}^2\_B$ given by


$$\mathcal{C}\_1 :=V(aX^2+bY^2+cZ^2+dXY +eXZ+fYZ) \subset \ \\\\\ \operatorname{Proj} (K[a,b,c,d,e,f,g,h,i,j,k,l][X,Y,Z]) = \mathbb{P}^2\_B $$


and similarly $\mathcal{C}\_2 :=V(gX^2+hY^2+iZ^2+jXY +kXZ+lYZ) \subset \ \mathbb{P}^2\_B $. Consider $\mathcal{C}\_1 \cap \mathcal{C}\_2 $.


**(a):** Show that the intersection $\mathcal{C}\_1 \cap \mathcal{C}\_2 $ is *generically reduced* by showing that the canonical induced projection


$$ p\_1:\mathcal{C}\_1 \cap \mathcal{C}\_2 \subset \mathbb{P}^2\_B \to B=\mathbb{A}^{12}\_K $$


has a fiber consisting of four distinct (hence reduced and K-rational) points.


It's easy to see that there exist such fiber with respect this projection map $p\_1$ which has this property. (consider the maximal ideal $(a-x\_a, b-x\_b,..., l-x\_l) $ with $x\_n \in K$ choosen general enough. Essentially that's Bezout's theorem.


So we can easily find such fibers, having obviously this property, expecially beeing reduced.


**Question:** But how can we conclude from this that $ \mathcal{C}\_1 \cap \mathcal{C}\_2 $ is generically reduced, ie that there exist an open (wlog affine) subscheme $U =\operatorname{Spec} R \subset \mathcal{C}\_1 \cap \mathcal{C}\_2 $ with reduced $R$.


**Metaquestion:** going back to the "motivation" at the beginning: Is there general "principle/ philosophy" behind such kind of argumentation techniques? For example, is there a "interesting" class of algebro geometric properties $\mathcal{P}$ known, for which such argumentation technique as in presented exercise go through, ie $f:X \to Y$ "fibration-like", $Y$ and some fiber $X\_y$ have property $\mathcal{P}$, then $X$ has it *generically* too?



Existence of a reduced fiber implies generically reduced (Exercise III-74 from Geometry of Schemes)
As noted by Jason Starr, the *Generic Principle* has been worked out in details by Grothendieck in EGA IV. If your French is not on top these days, the following version looks quite appealing (and can be found as Theorem 23.9 and 24.4 in Matsumura's *Commutative Ring Theory*) :


Let $f : X \longrightarrow Y$ be a **flat** (this is the expected fibration-like hypothesis), finite type morphism of Noetherian schemes with $X$ irreducible and let $y \in Y$.


$\bullet$ if $Y$ and $f^{-1}(y)$ satisfy the condition $R\_k$, then there is a dense open subset $X' \subset X$ which satisfies the condition $R\_k$.


$\bullet$ if $Y$ and $f^{-1}(y)$ satisfy the condition $S\_k$, then there is a dense open subset $X' \subset X$ which satisfies the condition $S\_k$.


Note that reduced is $R\_0$ + $S\_1$ (see for instane [Lemma 10.157.3 in *The Stack Project*](https://stacks.math.columbia.edu/tag/031R)), so that your expectation in the Metaquestion is correct.


