Advantages of normalization of varieties
$begingroup$
I am inexperienced in algebraic geometry, all I learned from this was reading some class notes and many questions clarified with the help of this platform. I'm reading a part of a book and I found some more doubts:
"Let $X subset mathbb{P}^n$ be an irreducible projective variety. Consider the Gauss map
$$
gamma: X_{reg}longrightarrow mathbb{G}(k,n)
$$
that assigns to each point $p in X_{reg}$ the translate to the origin of the projectivized tangent hyperplane $mathbb{P}T_p(X_{reg}).$ Let $Gamma$ be the closure of the graph of $gamma$ in $X timesmathbb{G}(k,n)$. Let $widetilde {Gamma}$ be the normalization of $Gamma$. We have a natural morphism $alpha:widetilde {Gamma} longrightarrow mathbb{G}(k,n)$ induced by
projection onto the second factor."
1) It is correct that $alpha(bigstar)=p_2 circ nu(bigstar)$, where $p_2$ is projection onto the second factor and $nu:widetilde{Gamma} longrightarrow
Gamma$ is regular map birational, given by normalization ???
2) What is the advantage of working with $alpha$ instead of directly $p_2$? What do you get by looking at $widetilde {Gamma}$ instead of $Gamma$???
Thanks in advance.
algebraic-geometry
asked Jan 14 at 1:02
ManoelManoel
942518
$endgroup$
add a comment |
$begingroup$
I am inexperienced in algebraic geometry, all I learned from this was reading some class notes and many questions clarified with the help of this platform. I'm reading a part of a book and I found some more doubts:
"Let $X subset mathbb{P}^n$ be an irreducible projective variety. Consider the Gauss map
$$
gamma: X_{reg}longrightarrow mathbb{G}(k,n)
$$
that assigns to each point $p in X_{reg}$ the translate to the origin of the projectivized tangent hyperplane $mathbb{P}T_p(X_{reg}).$ Let $Gamma$ be the closure of the graph of $gamma$ in $X timesmathbb{G}(k,n)$. Let $widetilde {Gamma}$ be the normalization of $Gamma$. We have a natural morphism $alpha:widetilde {Gamma} longrightarrow mathbb{G}(k,n)$ induced by
projection onto the second factor."
1) It is correct that $alpha(bigstar)=p_2 circ nu(bigstar)$, where $p_2$ is projection onto the second factor and $nu:widetilde{Gamma} longrightarrow
Gamma$ is regular map birational, given by normalization ???
2) What is the advantage of working with $alpha$ instead of directly $p_2$? What do you get by looking at $widetilde {Gamma}$ instead of $Gamma$???
Thanks in advance.
algebraic-geometry
asked Jan 14 at 1:02
ManoelManoel
942518
$endgroup$
add a comment |
$begingroup$
I am inexperienced in algebraic geometry, all I learned from this was reading some class notes and many questions clarified with the help of this platform. I'm reading a part of a book and I found some more doubts:
"Let $X subset mathbb{P}^n$ be an irreducible projective variety. Consider the Gauss map
$$
gamma: X_{reg}longrightarrow mathbb{G}(k,n)
$$
that assigns to each point $p in X_{reg}$ the translate to the origin of the projectivized tangent hyperplane $mathbb{P}T_p(X_{reg}).$ Let $Gamma$ be the closure of the graph of $gamma$ in $X timesmathbb{G}(k,n)$. Let $widetilde {Gamma}$ be the normalization of $Gamma$. We have a natural morphism $alpha:widetilde {Gamma} longrightarrow mathbb{G}(k,n)$ induced by
projection onto the second factor."
1) It is correct that $alpha(bigstar)=p_2 circ nu(bigstar)$, where $p_2$ is projection onto the second factor and $nu:widetilde{Gamma} longrightarrow
Gamma$ is regular map birational, given by normalization ???
2) What is the advantage of working with $alpha$ instead of directly $p_2$? What do you get by looking at $widetilde {Gamma}$ instead of $Gamma$???
Thanks in advance.
algebraic-geometry
asked Jan 14 at 1:02
ManoelManoel
942518
$endgroup$
I am inexperienced in algebraic geometry, all I learned from this was reading some class notes and many questions clarified with the help of this platform. I'm reading a part of a book and I found some more doubts:
"Let $X subset mathbb{P}^n$ be an irreducible projective variety. Consider the Gauss map
$$
gamma: X_{reg}longrightarrow mathbb{G}(k,n)
$$
that assigns to each point $p in X_{reg}$ the translate to the origin of the projectivized tangent hyperplane $mathbb{P}T_p(X_{reg}).$ Let $Gamma$ be the closure of the graph of $gamma$ in $X timesmathbb{G}(k,n)$. Let $widetilde {Gamma}$ be the normalization of $Gamma$. We have a natural morphism $alpha:widetilde {Gamma} longrightarrow mathbb{G}(k,n)$ induced by
projection onto the second factor."
1) It is correct that $alpha(bigstar)=p_2 circ nu(bigstar)$, where $p_2$ is projection onto the second factor and $nu:widetilde{Gamma} longrightarrow
Gamma$ is regular map birational, given by normalization ???
2) What is the advantage of working with $alpha$ instead of directly $p_2$? What do you get by looking at $widetilde {Gamma}$ instead of $Gamma$???
Thanks in advance.
algebraic-geometry
algebraic-geometry
asked Jan 14 at 1:02
ManoelManoel
942518
asked Jan 14 at 1:02
ManoelManoel
942518
asked Jan 14 at 1:02
ManoelManoel
942518
asked Jan 14 at 1:02
ManoelManoel
942518
asked Jan 14 at 1:02
ManoelManoel
942518
942518
add a comment |
add a comment |
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