Ordinary r-fold point on the dual curve
$begingroup$
Let $C$ a projective curve in $mathbb{P}^2(mathbb{C})$. We say a line $L subset mathbb{P}^2$ is mulltiple tangent of $C$ if there are $P_1, dots P_k$ points on $C$ such that $L$ is the tangent of $C$ at $P_i$ for every $i$ and $k geq 2$ and none of them is an inflection point.
We have the Gauss map: $nu :C to C^*$ from the curve to its dual.Now, we have $nu(P_i)=Q$ for every $i$. I would like to get a proof of the fact that $Q$ is an ordinary $k$ fold point of $C^*$.
I think that the fact that we are on $mathbb{C}$ and we have the analytic topology could be used to make things simpler, but I do not really understand how the dual curve is made.
EDIT: I tried to write an explicit parametrization of the dual curve, but I still can't conclude.(I'm gonna work over $mathbb{C}$ because I do not know anything about completion. Let's say for the sake of simplicity, $k=2$ and $P_1=[1 ;0 ;0],P_2=[0; 1 ; 0]$ and $L={z=0}$. Locally, near $P_1$ the curve has a parametrization of the kind $(x,z(x))$ in the standard affine coordinates, such that $z'(0)=0,z''(0) neq 0$. Now , the points in the tangent curve near the image of $P_1$ should be written as $[z'(x) ; -xz'(x)+z(x) ; 1]$. The problem is that I do not know how to find the tangents to the dual curve having written this.
algebraic-geometry riemann-surfaces projective-space
asked Jan 19 at 14:42
Tommaso ScognamiglioTommaso Scognamiglio
492312
$endgroup$
add a comment |
$begingroup$
Let $C$ a projective curve in $mathbb{P}^2(mathbb{C})$. We say a line $L subset mathbb{P}^2$ is mulltiple tangent of $C$ if there are $P_1, dots P_k$ points on $C$ such that $L$ is the tangent of $C$ at $P_i$ for every $i$ and $k geq 2$ and none of them is an inflection point.
We have the Gauss map: $nu :C to C^*$ from the curve to its dual.Now, we have $nu(P_i)=Q$ for every $i$. I would like to get a proof of the fact that $Q$ is an ordinary $k$ fold point of $C^*$.
I think that the fact that we are on $mathbb{C}$ and we have the analytic topology could be used to make things simpler, but I do not really understand how the dual curve is made.
EDIT: I tried to write an explicit parametrization of the dual curve, but I still can't conclude.(I'm gonna work over $mathbb{C}$ because I do not know anything about completion. Let's say for the sake of simplicity, $k=2$ and $P_1=[1 ;0 ;0],P_2=[0; 1 ; 0]$ and $L={z=0}$. Locally, near $P_1$ the curve has a parametrization of the kind $(x,z(x))$ in the standard affine coordinates, such that $z'(0)=0,z''(0) neq 0$. Now , the points in the tangent curve near the image of $P_1$ should be written as $[z'(x) ; -xz'(x)+z(x) ; 1]$. The problem is that I do not know how to find the tangents to the dual curve having written this.
algebraic-geometry riemann-surfaces projective-space
asked Jan 19 at 14:42
Tommaso ScognamiglioTommaso Scognamiglio
492312
$endgroup$
$begingroup$
This is not true unless you require that the curve has no inflection points (consider $V(y-x^3)$, for instance). And the analytic topology is unnecessary - one can do everything in completions and it will work just fine for any field of characteristic not $2$. The first thing to do is to get a handle on how the dual curve works: you may wish to peruse the wikipedia page about it, for instance. From there, look at what happens to the equation of the tangent line near $P_i$ that map to the same $Q$ - since it's not an inflection point, there's a specific relation that must be satisfied...
$endgroup$
– KReiser
Jan 20 at 8:06
$begingroup$
I edited for what about inflection points thank you
$endgroup$
– Tommaso Scognamiglio
Jan 20 at 8:19
add a comment |
$begingroup$
Let $C$ a projective curve in $mathbb{P}^2(mathbb{C})$. We say a line $L subset mathbb{P}^2$ is mulltiple tangent of $C$ if there are $P_1, dots P_k$ points on $C$ such that $L$ is the tangent of $C$ at $P_i$ for every $i$ and $k geq 2$ and none of them is an inflection point.
We have the Gauss map: $nu :C to C^*$ from the curve to its dual.Now, we have $nu(P_i)=Q$ for every $i$. I would like to get a proof of the fact that $Q$ is an ordinary $k$ fold point of $C^*$.
I think that the fact that we are on $mathbb{C}$ and we have the analytic topology could be used to make things simpler, but I do not really understand how the dual curve is made.
EDIT: I tried to write an explicit parametrization of the dual curve, but I still can't conclude.(I'm gonna work over $mathbb{C}$ because I do not know anything about completion. Let's say for the sake of simplicity, $k=2$ and $P_1=[1 ;0 ;0],P_2=[0; 1 ; 0]$ and $L={z=0}$. Locally, near $P_1$ the curve has a parametrization of the kind $(x,z(x))$ in the standard affine coordinates, such that $z'(0)=0,z''(0) neq 0$. Now , the points in the tangent curve near the image of $P_1$ should be written as $[z'(x) ; -xz'(x)+z(x) ; 1]$. The problem is that I do not know how to find the tangents to the dual curve having written this.
algebraic-geometry riemann-surfaces projective-space
asked Jan 19 at 14:42
Tommaso ScognamiglioTommaso Scognamiglio
492312
$endgroup$
Let $C$ a projective curve in $mathbb{P}^2(mathbb{C})$. We say a line $L subset mathbb{P}^2$ is mulltiple tangent of $C$ if there are $P_1, dots P_k$ points on $C$ such that $L$ is the tangent of $C$ at $P_i$ for every $i$ and $k geq 2$ and none of them is an inflection point.
We have the Gauss map: $nu :C to C^*$ from the curve to its dual.Now, we have $nu(P_i)=Q$ for every $i$. I would like to get a proof of the fact that $Q$ is an ordinary $k$ fold point of $C^*$.
I think that the fact that we are on $mathbb{C}$ and we have the analytic topology could be used to make things simpler, but I do not really understand how the dual curve is made.
EDIT: I tried to write an explicit parametrization of the dual curve, but I still can't conclude.(I'm gonna work over $mathbb{C}$ because I do not know anything about completion. Let's say for the sake of simplicity, $k=2$ and $P_1=[1 ;0 ;0],P_2=[0; 1 ; 0]$ and $L={z=0}$. Locally, near $P_1$ the curve has a parametrization of the kind $(x,z(x))$ in the standard affine coordinates, such that $z'(0)=0,z''(0) neq 0$. Now , the points in the tangent curve near the image of $P_1$ should be written as $[z'(x) ; -xz'(x)+z(x) ; 1]$. The problem is that I do not know how to find the tangents to the dual curve having written this.
algebraic-geometry riemann-surfaces projective-space
algebraic-geometry riemann-surfaces projective-space
asked Jan 19 at 14:42
Tommaso ScognamiglioTommaso Scognamiglio
492312
asked Jan 19 at 14:42
Tommaso ScognamiglioTommaso Scognamiglio
492312
edited Jan 20 at 9:52
Tommaso Scognamiglio
asked Jan 19 at 14:42
Tommaso ScognamiglioTommaso Scognamiglio
492312
asked Jan 19 at 14:42
Tommaso ScognamiglioTommaso Scognamiglio
492312
asked Jan 19 at 14:42
Tommaso ScognamiglioTommaso Scognamiglio
492312
492312
$begingroup$
This is not true unless you require that the curve has no inflection points (consider $V(y-x^3)$, for instance). And the analytic topology is unnecessary - one can do everything in completions and it will work just fine for any field of characteristic not $2$. The first thing to do is to get a handle on how the dual curve works: you may wish to peruse the wikipedia page about it, for instance. From there, look at what happens to the equation of the tangent line near $P_i$ that map to the same $Q$ - since it's not an inflection point, there's a specific relation that must be satisfied...
$endgroup$
– KReiser
Jan 20 at 8:06
$begingroup$
I edited for what about inflection points thank you
$endgroup$
– Tommaso Scognamiglio
Jan 20 at 8:19
add a comment |
$begingroup$
This is not true unless you require that the curve has no inflection points (consider $V(y-x^3)$, for instance). And the analytic topology is unnecessary - one can do everything in completions and it will work just fine for any field of characteristic not $2$. The first thing to do is to get a handle on how the dual curve works: you may wish to peruse the wikipedia page about it, for instance. From there, look at what happens to the equation of the tangent line near $P_i$ that map to the same $Q$ - since it's not an inflection point, there's a specific relation that must be satisfied...
$endgroup$
– KReiser
Jan 20 at 8:06
$begingroup$
I edited for what about inflection points thank you
$endgroup$
– Tommaso Scognamiglio
Jan 20 at 8:19
$begingroup$
This is not true unless you require that the curve has no inflection points (consider $V(y-x^3)$, for instance). And the analytic topology is unnecessary - one can do everything in completions and it will work just fine for any field of characteristic not $2$. The first thing to do is to get a handle on how the dual curve works: you may wish to peruse the wikipedia page about it, for instance. From there, look at what happens to the equation of the tangent line near $P_i$ that map to the same $Q$ - since it's not an inflection point, there's a specific relation that must be satisfied...
$endgroup$
– KReiser
Jan 20 at 8:06
$begingroup$
This is not true unless you require that the curve has no inflection points (consider $V(y-x^3)$, for instance). And the analytic topology is unnecessary - one can do everything in completions and it will work just fine for any field of characteristic not $2$. The first thing to do is to get a handle on how the dual curve works: you may wish to peruse the wikipedia page about it, for instance. From there, look at what happens to the equation of the tangent line near $P_i$ that map to the same $Q$ - since it's not an inflection point, there's a specific relation that must be satisfied...
$endgroup$
– KReiser
Jan 20 at 8:06
$begingroup$
I edited for what about inflection points thank you
$endgroup$
– Tommaso Scognamiglio
Jan 20 at 8:19
$begingroup$
I edited for what about inflection points thank you
$endgroup$
– Tommaso Scognamiglio
Jan 20 at 8:19
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%2f3079417%2fordinary-r-fold-point-on-the-dual-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%2f3079417%2fordinary-r-fold-point-on-the-dual-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
$begingroup$
This is not true unless you require that the curve has no inflection points (consider $V(y-x^3)$, for instance). And the analytic topology is unnecessary - one can do everything in completions and it will work just fine for any field of characteristic not $2$. The first thing to do is to get a handle on how the dual curve works: you may wish to peruse the wikipedia page about it, for instance. From there, look at what happens to the equation of the tangent line near $P_i$ that map to the same $Q$ - since it's not an inflection point, there's a specific relation that must be satisfied...
$endgroup$
– KReiser
Jan 20 at 8:06
$begingroup$
I edited for what about inflection points thank you
$endgroup$
– Tommaso Scognamiglio
Jan 20 at 8:19