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This question deals with a concrete exercise from Geomerty of Schemes 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 (sloppy said something "fibration like") $f:X \to Y$ 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?

This question deals with a concrete exercise from Geomerty of Schemes 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?

This question deals with a concrete exercise from Geomerty of Schemes 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 (sloppy I mean flat, ie "fibration like") $f:X \to Y$ 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?

This question deals with a concrete exercise from Geomerty of Schemes 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 (sloppy said something "fibration like") $f:X \to Y$ 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?

This question deals with a concrete exercise from Geomerty of Schemes 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 (sloppy I mean flat, ie "fibration like") $f:X \to Y$ of schemes over base field $K$, assume $Y$ (wlog assume it to be affine) has some property $\mathcal{P}$ and say we have additional assumpion that the 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 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?

This question deals with a concrete exercise from Geomerty of Schemes 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 (sloppy I mean flat, ie "fibration like") $f:X \to Y$ 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?