Dgdrxrt

Should I cover my bicycle overnight while bikepacking?

Do UK voters know if their MP will be the Speaker of the House?

Watching something be piped to a file live with tail

How do conventional missiles fly?

What killed these X2 caps?

Size of subfigure fitting its content (tikzpicture)

Forgetting the musical notes while performing in concert

Is there a hemisphere-neutral way of specifying a season?

How do I gain back my faith in my PhD degree?

Why was the shrinking from 8″ made only to 5.25″ and not smaller (4″ or less)?

How to show a landlord what we have in savings?

How does a predictive coding aid in lossless compression?

Examples of smooth manifolds admitting inbetween one and a continuum of complex structures

Is there an expression that means doing something right before you will need it rather than doing it in case you might need it?

Could the museum Saturn V's be refitted for one more flight?

Solving a recurrence relation (poker chips)

What do you call someone who asks many questions?

Can a virus destroy the BIOS of a modern computer?

How to Recreate this in LaTeX? (Unsure What the Notation is Called)

Short story with a alien planet, government officials must wear exploding medallions

Do scales need to be in alphabetical order?

Alternative to sending password over mail?

Why no variance term in Bayesian logistic regression?

If human space travel is limited by the G force vulnerability, is there a way to counter G forces?

space of projective plane curves of degree d

Genus formula for curve on surfaceBirational Maps of Nonsingular Projective CurvesGeometric interpretation of the Riemann-Roch for curvesDegree of ample bundle over projective curve is positiveProjective curve $x^d+y^d+z^d=0$ is nonsingular using Jacobian matrixResolving a node singularity on a plane curve“Canonical map” of singular stable curvesWhy geometrically irreducible quartic curves in projective plane corresponds to an open subset of projective space of dimension 14Only finitely many genus $g$ smooth projective curves over a finite fieldRemoving singular points from curves

0

$begingroup$

This is basically from Hartshorne Exercise I.5.13 where he writes there is a correspondence from the set of plane projective curves of degree $d$ to points in a projective space $mathbb P^N$ where $N=binomd+22-1$. All seems reasonable and I can easily produce this canonical correspondence.

However he writes that this is not 1-1 if we look at the image of a curve of degree $d$ that is reducible. I seem to be too stupid to find two points in the preimage of a point in $mathbb P^N$. Perhaps I misunderstood something?

algebraic-geometry algebraic-curves moduli-space

share|cite|improve this question

edited Mar 28 at 20:54

quantum

asked Mar 28 at 20:30

quantumquantum

538210

$endgroup$

add a comment | 

0

$begingroup$

This is basically from Hartshorne Exercise I.5.13 where he writes there is a correspondence from the set of plane projective curves of degree $d$ to points in a projective space $mathbb P^N$ where $N=binomd+22-1$. All seems reasonable and I can easily produce this canonical correspondence.

However he writes that this is not 1-1 if we look at the image of a curve of degree $d$ that is reducible. I seem to be too stupid to find two points in the preimage of a point in $mathbb P^N$. Perhaps I misunderstood something?

algebraic-geometry algebraic-curves moduli-space

share|cite|improve this question

edited Mar 28 at 20:54

quantum

asked Mar 28 at 20:30

quantumquantum

538210

$endgroup$

add a comment | 

$begingroup$

This is basically from Hartshorne Exercise I.5.13 where he writes there is a correspondence from the set of plane projective curves of degree $d$ to points in a projective space $mathbb P^N$ where $N=binomd+22-1$. All seems reasonable and I can easily produce this canonical correspondence.

However he writes that this is not 1-1 if we look at the image of a curve of degree $d$ that is reducible. I seem to be too stupid to find two points in the preimage of a point in $mathbb P^N$. Perhaps I misunderstood something?

algebraic-geometry algebraic-curves moduli-space

share|cite|improve this question

edited Mar 28 at 20:54

quantum

asked Mar 28 at 20:30

quantumquantum

538210

$endgroup$

share|cite|improve this question

edited Mar 28 at 20:54

quantum

asked Mar 28 at 20:30

quantumquantum

538210

share|cite|improve this question

edited Mar 28 at 20:54

quantum

asked Mar 28 at 20:30

quantumquantum

538210

asked Mar 28 at 20:30

quantumquantum

538210

asked Mar 28 at 20:30

quantumquantum

538210

1 Answer
1

active

oldest

votes

0

$begingroup$

$newcommandPmathbb P$I think I know why I was not getting the example:

1. I was thinking of a function from the set of projective plane curves of degree $d$ to $P^N$
   


2. I thought I could get an example if $d=2$ (maybe this is not posssible).

So Hartshorne meant a map from points in $P^N$ to the set of curves that can be defined by forms of degree $d$. So not only reverse of what I thought, but also we are not talking about the degree of the curve here which I misunderstood as well. One way to get a fiber with more than one point using this map is to consider $d=4$ and now the point in $P^N$ corresponding to $x^2y^2$ which is different from the point in $P^N$ corresponding to $x^3y$ map to the same curve.

share|cite|improve this answer

answered Mar 28 at 20:54

quantumquantum

538210

$endgroup$

add a comment | 

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
);



);

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3166383%2fspace-of-projective-plane-curves-of-degree-d%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

0

$begingroup$

$newcommandPmathbb P$I think I know why I was not getting the example:

1. I was thinking of a function from the set of projective plane curves of degree $d$ to $P^N$
   


2. I thought I could get an example if $d=2$ (maybe this is not posssible).

So Hartshorne meant a map from points in $P^N$ to the set of curves that can be defined by forms of degree $d$. So not only reverse of what I thought, but also we are not talking about the degree of the curve here which I misunderstood as well. One way to get a fiber with more than one point using this map is to consider $d=4$ and now the point in $P^N$ corresponding to $x^2y^2$ which is different from the point in $P^N$ corresponding to $x^3y$ map to the same curve.

share|cite|improve this answer

answered Mar 28 at 20:54

quantumquantum

538210

$endgroup$

add a comment | 

0

$begingroup$

$newcommandPmathbb P$I think I know why I was not getting the example:

1. I was thinking of a function from the set of projective plane curves of degree $d$ to $P^N$
   


2. I thought I could get an example if $d=2$ (maybe this is not posssible).

So Hartshorne meant a map from points in $P^N$ to the set of curves that can be defined by forms of degree $d$. So not only reverse of what I thought, but also we are not talking about the degree of the curve here which I misunderstood as well. One way to get a fiber with more than one point using this map is to consider $d=4$ and now the point in $P^N$ corresponding to $x^2y^2$ which is different from the point in $P^N$ corresponding to $x^3y$ map to the same curve.

share|cite|improve this answer

answered Mar 28 at 20:54

quantumquantum

538210

$endgroup$

add a comment | 

$begingroup$

$newcommandPmathbb P$I think I know why I was not getting the example:

1. I was thinking of a function from the set of projective plane curves of degree $d$ to $P^N$
   


2. I thought I could get an example if $d=2$ (maybe this is not posssible).

So Hartshorne meant a map from points in $P^N$ to the set of curves that can be defined by forms of degree $d$. So not only reverse of what I thought, but also we are not talking about the degree of the curve here which I misunderstood as well. One way to get a fiber with more than one point using this map is to consider $d=4$ and now the point in $P^N$ corresponding to $x^2y^2$ which is different from the point in $P^N$ corresponding to $x^3y$ map to the same curve.

share|cite|improve this answer

answered Mar 28 at 20:54

quantumquantum

538210

$endgroup$

share|cite|improve this answer

answered Mar 28 at 20:54

quantumquantum

538210

answered Mar 28 at 20:54

quantumquantum

538210

answered Mar 28 at 20:54

quantumquantum

538210