Genus of a smooth projective curve
$begingroup$
I was trying to prove that the genus of a smooth projective complex curve $F=0$ of degree $d$ is $(d-1)(d-2)/2$.
My attempt was to take the standard projection $$pi:mathbb{P}^2 to mathbb{P}^1$$ $$[x ;y ; z] to [x ; z]$$ which has degree $d$ and then apply Riemann Hurwitz formula.
The problem is that I do not know how to explicitly write the multiplictity of this map at the ramification points(which are the ones for which $dfrac{partial F}{partial y}=0$). In particular, I would like to say that the multiplicity of the map $pi $ in a point $x in mathbb{P}^2$ is $Ileft(x,F cap dfrac{partial F}{partial y}+1right)$, but I'm stuck (the last I is the intersection multiplicity of the two algebraic curves).
algebraic-geometry complex-geometry algebraic-curves riemann-surfaces
asked Jan 16 at 18:16
Tommaso ScognamiglioTommaso Scognamiglio
581412
$endgroup$
add a comment |
$begingroup$
I was trying to prove that the genus of a smooth projective complex curve $F=0$ of degree $d$ is $(d-1)(d-2)/2$.
My attempt was to take the standard projection $$pi:mathbb{P}^2 to mathbb{P}^1$$ $$[x ;y ; z] to [x ; z]$$ which has degree $d$ and then apply Riemann Hurwitz formula.
The problem is that I do not know how to explicitly write the multiplictity of this map at the ramification points(which are the ones for which $dfrac{partial F}{partial y}=0$). In particular, I would like to say that the multiplicity of the map $pi $ in a point $x in mathbb{P}^2$ is $Ileft(x,F cap dfrac{partial F}{partial y}+1right)$, but I'm stuck (the last I is the intersection multiplicity of the two algebraic curves).
algebraic-geometry complex-geometry algebraic-curves riemann-surfaces
asked Jan 16 at 18:16
Tommaso ScognamiglioTommaso Scognamiglio
581412
$endgroup$
$begingroup$
Are you asking about how to finish this particular proof using the Riemann-Hurwitz formula, or are you interested in alternate ways to prove this statement?
$endgroup$
– KReiser
Jan 16 at 21:00
$begingroup$
I would like to see a proof using Riemann Hurwitz formula
$endgroup$
– Tommaso Scognamiglio
Jan 16 at 21:44
$begingroup$
Is it okay to use adjunction + cohomology ? Then $K_C = mathcal{O}_C(n-3) $. Since the curve is defined by a single equation of degree d it's easy to see that $K_C$ has exactly the same global sections as on $P^2$.
$endgroup$
– aginensky
Jan 16 at 22:52
1
$begingroup$
I think this is basically a duplicated of this question.
$endgroup$
– André 3000
Jan 17 at 4:17
$begingroup$
@André3000 Tthe answer in there is just a broken link, it might be worth it having a written answer.
$endgroup$
– user347489
Jan 17 at 8:31
add a comment |
$begingroup$
I was trying to prove that the genus of a smooth projective complex curve $F=0$ of degree $d$ is $(d-1)(d-2)/2$.
My attempt was to take the standard projection $$pi:mathbb{P}^2 to mathbb{P}^1$$ $$[x ;y ; z] to [x ; z]$$ which has degree $d$ and then apply Riemann Hurwitz formula.
The problem is that I do not know how to explicitly write the multiplictity of this map at the ramification points(which are the ones for which $dfrac{partial F}{partial y}=0$). In particular, I would like to say that the multiplicity of the map $pi $ in a point $x in mathbb{P}^2$ is $Ileft(x,F cap dfrac{partial F}{partial y}+1right)$, but I'm stuck (the last I is the intersection multiplicity of the two algebraic curves).
algebraic-geometry complex-geometry algebraic-curves riemann-surfaces
asked Jan 16 at 18:16
Tommaso ScognamiglioTommaso Scognamiglio
581412
$endgroup$
I was trying to prove that the genus of a smooth projective complex curve $F=0$ of degree $d$ is $(d-1)(d-2)/2$.
My attempt was to take the standard projection $$pi:mathbb{P}^2 to mathbb{P}^1$$ $$[x ;y ; z] to [x ; z]$$ which has degree $d$ and then apply Riemann Hurwitz formula.
The problem is that I do not know how to explicitly write the multiplictity of this map at the ramification points(which are the ones for which $dfrac{partial F}{partial y}=0$). In particular, I would like to say that the multiplicity of the map $pi $ in a point $x in mathbb{P}^2$ is $Ileft(x,F cap dfrac{partial F}{partial y}+1right)$, but I'm stuck (the last I is the intersection multiplicity of the two algebraic curves).
algebraic-geometry complex-geometry algebraic-curves riemann-surfaces
algebraic-geometry complex-geometry algebraic-curves riemann-surfaces
asked Jan 16 at 18:16
Tommaso ScognamiglioTommaso Scognamiglio
581412
asked Jan 16 at 18:16
Tommaso ScognamiglioTommaso Scognamiglio
581412
asked Jan 16 at 18:16
Tommaso ScognamiglioTommaso Scognamiglio
581412
asked Jan 16 at 18:16
Tommaso ScognamiglioTommaso Scognamiglio
581412
asked Jan 16 at 18:16
Tommaso ScognamiglioTommaso Scognamiglio
581412
581412
$begingroup$
Are you asking about how to finish this particular proof using the Riemann-Hurwitz formula, or are you interested in alternate ways to prove this statement?
$endgroup$
– KReiser
Jan 16 at 21:00
$begingroup$
I would like to see a proof using Riemann Hurwitz formula
$endgroup$
– Tommaso Scognamiglio
Jan 16 at 21:44
$begingroup$
Is it okay to use adjunction + cohomology ? Then $K_C = mathcal{O}_C(n-3) $. Since the curve is defined by a single equation of degree d it's easy to see that $K_C$ has exactly the same global sections as on $P^2$.
$endgroup$
– aginensky
Jan 16 at 22:52
1
$begingroup$
I think this is basically a duplicated of this question.
$endgroup$
– André 3000
Jan 17 at 4:17
$begingroup$
@André3000 Tthe answer in there is just a broken link, it might be worth it having a written answer.
$endgroup$
– user347489
Jan 17 at 8:31
add a comment |
$begingroup$
Are you asking about how to finish this particular proof using the Riemann-Hurwitz formula, or are you interested in alternate ways to prove this statement?
$endgroup$
– KReiser
Jan 16 at 21:00
$begingroup$
I would like to see a proof using Riemann Hurwitz formula
$endgroup$
– Tommaso Scognamiglio
Jan 16 at 21:44
$begingroup$
Is it okay to use adjunction + cohomology ? Then $K_C = mathcal{O}_C(n-3) $. Since the curve is defined by a single equation of degree d it's easy to see that $K_C$ has exactly the same global sections as on $P^2$.
$endgroup$
– aginensky
Jan 16 at 22:52
1
$begingroup$
I think this is basically a duplicated of this question.
$endgroup$
– André 3000
Jan 17 at 4:17
$begingroup$
@André3000 Tthe answer in there is just a broken link, it might be worth it having a written answer.
$endgroup$
– user347489
Jan 17 at 8:31
$begingroup$
Are you asking about how to finish this particular proof using the Riemann-Hurwitz formula, or are you interested in alternate ways to prove this statement?
$endgroup$
– KReiser
Jan 16 at 21:00
$begingroup$
Are you asking about how to finish this particular proof using the Riemann-Hurwitz formula, or are you interested in alternate ways to prove this statement?
$endgroup$
– KReiser
Jan 16 at 21:00
$begingroup$
I would like to see a proof using Riemann Hurwitz formula
$endgroup$
– Tommaso Scognamiglio
Jan 16 at 21:44
$begingroup$
I would like to see a proof using Riemann Hurwitz formula
$endgroup$
– Tommaso Scognamiglio
Jan 16 at 21:44
$begingroup$
Is it okay to use adjunction + cohomology ? Then $K_C = mathcal{O}_C(n-3) $. Since the curve is defined by a single equation of degree d it's easy to see that $K_C$ has exactly the same global sections as on $P^2$.
$endgroup$
– aginensky
Jan 16 at 22:52
$begingroup$
Is it okay to use adjunction + cohomology ? Then $K_C = mathcal{O}_C(n-3) $. Since the curve is defined by a single equation of degree d it's easy to see that $K_C$ has exactly the same global sections as on $P^2$.
$endgroup$
– aginensky
Jan 16 at 22:52
1
1
$begingroup$
I think this is basically a duplicated of this question.
$endgroup$
– André 3000
Jan 17 at 4:17
$begingroup$
I think this is basically a duplicated of this question.
$endgroup$
– André 3000
Jan 17 at 4:17
$begingroup$
@André3000 Tthe answer in there is just a broken link, it might be worth it having a written answer.
$endgroup$
– user347489
Jan 17 at 8:31
$begingroup$
@André3000 Tthe answer in there is just a broken link, it might be worth it having a written answer.
$endgroup$
– user347489
Jan 17 at 8:31
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3076083%2fgenus-of-a-smooth-projective-curve%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3076083%2fgenus-of-a-smooth-projective-curve%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
Are you asking about how to finish this particular proof using the Riemann-Hurwitz formula, or are you interested in alternate ways to prove this statement?
$endgroup$
– KReiser
Jan 16 at 21:00
$begingroup$
I would like to see a proof using Riemann Hurwitz formula
$endgroup$
– Tommaso Scognamiglio
Jan 16 at 21:44
$begingroup$
Is it okay to use adjunction + cohomology ? Then $K_C = mathcal{O}_C(n-3) $. Since the curve is defined by a single equation of degree d it's easy to see that $K_C$ has exactly the same global sections as on $P^2$.
$endgroup$
– aginensky
Jan 16 at 22:52
1
$begingroup$
I think this is basically a duplicated of this question.
$endgroup$
– André 3000
Jan 17 at 4:17
$begingroup$
@André3000 Tthe answer in there is just a broken link, it might be worth it having a written answer.
$endgroup$
– user347489
Jan 17 at 8:31