Mixing
Glueing construction for projective scheme
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
I am self-studying Algebraic Geometry, mainly from Goertz/Wedhorn and the notes of Ravi Vakil, for my private interest. This is my first question on MSE (great site!).
I am trying to apply the glueing construction as described in Vakil (4.4.9) to his Exercise 4.5.A (in using slightly different notation):
Let
$$V_0=mathrmSpec k[x_1/x_0, x_2/x_0]/(1+(x_1/x_0)^2-(x_2/x_0)^2)=mathrmSpec A_0,$$
$$V_1=mathrmSpec k[x_0/x_1, x_2/x_1]/((x_0/x_1)^2 + 1 -(x_2/x_1)^2)=mathrmSpec A_1,$$
$$V_2=mathrmSpec k[x_0/x_2, x_1/x_2]/((x_0/x_2)^2+(x_1/x_2)^2-1)=mathrmSpec A_2.$$
And let $V_ij=mathrmSpec A_i[(x_j/x_i)^-1]=mathrmSpec A_ij, ineq j. $
I would like to see, that I can identify $V_ij=V_ji$ through $x_k/x_i=x_k/x_j x_j/x_i$ and $x_j/x_i=x_i/x_j (kneq i,j).$
If I apply this to $V_12$, the ideal $((x_0/x_1)^2 + 1 -(x_2/x_1)^2)$ in $A_12$ maps to $((x_0/x_2)^2 (x_2/x_1)^2+ 1- (x_1/x_2)^2)$ in $A_21$ and should be equal to $((x_0/x_2)^2+(x_1/x_2)^2-1)$.
I am unsure whether my reasoning so far is valid and if so, I cannot quite see why the two ideals are equal. Apologies, if this is a lengthy description for a trivial question.
algebraic-geometry projective-space
asked Aug 6 at 5:47
Jan
62
add a comment |Â
up vote
1
down vote
favorite
I am self-studying Algebraic Geometry, mainly from Goertz/Wedhorn and the notes of Ravi Vakil, for my private interest. This is my first question on MSE (great site!).
I am trying to apply the glueing construction as described in Vakil (4.4.9) to his Exercise 4.5.A (in using slightly different notation):
Let
$$V_0=mathrmSpec k[x_1/x_0, x_2/x_0]/(1+(x_1/x_0)^2-(x_2/x_0)^2)=mathrmSpec A_0,$$
$$V_1=mathrmSpec k[x_0/x_1, x_2/x_1]/((x_0/x_1)^2 + 1 -(x_2/x_1)^2)=mathrmSpec A_1,$$
$$V_2=mathrmSpec k[x_0/x_2, x_1/x_2]/((x_0/x_2)^2+(x_1/x_2)^2-1)=mathrmSpec A_2.$$
And let $V_ij=mathrmSpec A_i[(x_j/x_i)^-1]=mathrmSpec A_ij, ineq j. $
I would like to see, that I can identify $V_ij=V_ji$ through $x_k/x_i=x_k/x_j x_j/x_i$ and $x_j/x_i=x_i/x_j (kneq i,j).$
If I apply this to $V_12$, the ideal $((x_0/x_1)^2 + 1 -(x_2/x_1)^2)$ in $A_12$ maps to $((x_0/x_2)^2 (x_2/x_1)^2+ 1- (x_1/x_2)^2)$ in $A_21$ and should be equal to $((x_0/x_2)^2+(x_1/x_2)^2-1)$.
I am unsure whether my reasoning so far is valid and if so, I cannot quite see why the two ideals are equal. Apologies, if this is a lengthy description for a trivial question.
algebraic-geometry projective-space
asked Aug 6 at 5:47
Jan
62
Your calculation is not correct. To do this more clearly, write $A_12$ and $A_21$ as quotients of $k[x_0/x_2,x_1/x_2,x_2/x_1]$.
– user501746
Aug 6 at 9:42
@user501746 Thank you for your comment. I realised a little mistake in the definition of $V_ij$ and reversed the indices to $(x_j/x_i)^-1$. But I do not see where my calculation is wrong. Could you be more specific please? Thank you.
– Jan
Aug 6 at 12:22
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am self-studying Algebraic Geometry, mainly from Goertz/Wedhorn and the notes of Ravi Vakil, for my private interest. This is my first question on MSE (great site!).
I am trying to apply the glueing construction as described in Vakil (4.4.9) to his Exercise 4.5.A (in using slightly different notation):
Let
$$V_0=mathrmSpec k[x_1/x_0, x_2/x_0]/(1+(x_1/x_0)^2-(x_2/x_0)^2)=mathrmSpec A_0,$$
$$V_1=mathrmSpec k[x_0/x_1, x_2/x_1]/((x_0/x_1)^2 + 1 -(x_2/x_1)^2)=mathrmSpec A_1,$$
$$V_2=mathrmSpec k[x_0/x_2, x_1/x_2]/((x_0/x_2)^2+(x_1/x_2)^2-1)=mathrmSpec A_2.$$
And let $V_ij=mathrmSpec A_i[(x_j/x_i)^-1]=mathrmSpec A_ij, ineq j. $
I would like to see, that I can identify $V_ij=V_ji$ through $x_k/x_i=x_k/x_j x_j/x_i$ and $x_j/x_i=x_i/x_j (kneq i,j).$
If I apply this to $V_12$, the ideal $((x_0/x_1)^2 + 1 -(x_2/x_1)^2)$ in $A_12$ maps to $((x_0/x_2)^2 (x_2/x_1)^2+ 1- (x_1/x_2)^2)$ in $A_21$ and should be equal to $((x_0/x_2)^2+(x_1/x_2)^2-1)$.
I am unsure whether my reasoning so far is valid and if so, I cannot quite see why the two ideals are equal. Apologies, if this is a lengthy description for a trivial question.
algebraic-geometry projective-space
asked Aug 6 at 5:47
Jan
62
I am self-studying Algebraic Geometry, mainly from Goertz/Wedhorn and the notes of Ravi Vakil, for my private interest. This is my first question on MSE (great site!).
I am trying to apply the glueing construction as described in Vakil (4.4.9) to his Exercise 4.5.A (in using slightly different notation):
Let
$$V_0=mathrmSpec k[x_1/x_0, x_2/x_0]/(1+(x_1/x_0)^2-(x_2/x_0)^2)=mathrmSpec A_0,$$
$$V_1=mathrmSpec k[x_0/x_1, x_2/x_1]/((x_0/x_1)^2 + 1 -(x_2/x_1)^2)=mathrmSpec A_1,$$
$$V_2=mathrmSpec k[x_0/x_2, x_1/x_2]/((x_0/x_2)^2+(x_1/x_2)^2-1)=mathrmSpec A_2.$$
And let $V_ij=mathrmSpec A_i[(x_j/x_i)^-1]=mathrmSpec A_ij, ineq j. $
I would like to see, that I can identify $V_ij=V_ji$ through $x_k/x_i=x_k/x_j x_j/x_i$ and $x_j/x_i=x_i/x_j (kneq i,j).$
If I apply this to $V_12$, the ideal $((x_0/x_1)^2 + 1 -(x_2/x_1)^2)$ in $A_12$ maps to $((x_0/x_2)^2 (x_2/x_1)^2+ 1- (x_1/x_2)^2)$ in $A_21$ and should be equal to $((x_0/x_2)^2+(x_1/x_2)^2-1)$.
I am unsure whether my reasoning so far is valid and if so, I cannot quite see why the two ideals are equal. Apologies, if this is a lengthy description for a trivial question.
algebraic-geometry projective-space
asked Aug 6 at 5:47
Jan
62
edited Aug 6 at 12:15
asked Aug 6 at 5:47
Jan
62
asked Aug 6 at 5:47
Jan
62
asked Aug 6 at 5:47
Jan
62
62
Your calculation is not correct. To do this more clearly, write $A_12$ and $A_21$ as quotients of $k[x_0/x_2,x_1/x_2,x_2/x_1]$.
– user501746
Aug 6 at 9:42
@user501746 Thank you for your comment. I realised a little mistake in the definition of $V_ij$ and reversed the indices to $(x_j/x_i)^-1$. But I do not see where my calculation is wrong. Could you be more specific please? Thank you.
– Jan
Aug 6 at 12:22
add a comment |Â
Your calculation is not correct. To do this more clearly, write $A_12$ and $A_21$ as quotients of $k[x_0/x_2,x_1/x_2,x_2/x_1]$.
– user501746
Aug 6 at 9:42
@user501746 Thank you for your comment. I realised a little mistake in the definition of $V_ij$ and reversed the indices to $(x_j/x_i)^-1$. But I do not see where my calculation is wrong. Could you be more specific please? Thank you.
– Jan
Aug 6 at 12:22
Your calculation is not correct. To do this more clearly, write $A_12$ and $A_21$ as quotients of $k[x_0/x_2,x_1/x_2,x_2/x_1]$.
– user501746
Aug 6 at 9:42
Your calculation is not correct. To do this more clearly, write $A_12$ and $A_21$ as quotients of $k[x_0/x_2,x_1/x_2,x_2/x_1]$.
– user501746
Aug 6 at 9:42
@user501746 Thank you for your comment. I realised a little mistake in the definition of $V_ij$ and reversed the indices to $(x_j/x_i)^-1$. But I do not see where my calculation is wrong. Could you be more specific please? Thank you.
– Jan
Aug 6 at 12:22
@user501746 Thank you for your comment. I realised a little mistake in the definition of $V_ij$ and reversed the indices to $(x_j/x_i)^-1$. But I do not see where my calculation is wrong. Could you be more specific please? Thank you.
– Jan
Aug 6 at 12:22
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2873611%2fglueing-construction-for-projective-scheme%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Your calculation is not correct. To do this more clearly, write $A_12$ and $A_21$ as quotients of $k[x_0/x_2,x_1/x_2,x_2/x_1]$.
– user501746
Aug 6 at 9:42
@user501746 Thank you for your comment. I realised a little mistake in the definition of $V_ij$ and reversed the indices to $(x_j/x_i)^-1$. But I do not see where my calculation is wrong. Could you be more specific please? Thank you.
– Jan
Aug 6 at 12:22