What does dense mean in algebraic geometry?Relation between varieties in the sense of Serre's FAC and...
When were female captains banned from Starfleet?
Why Shazam when there is already Superman?
Problem with TransformedDistribution
How to implement a feedback to keep the DC gain at zero for this conceptual passive filter?
What is the evidence for the "tyranny of the majority problem" in a direct democracy context?
How can "mimic phobia" be cured or prevented?
What was the exact wording from Ivanhoe of this advice on how to free yourself from slavery?
Is there a single word describing earning money through any means?
How much character growth crosses the line into breaking the character
Creature in Shazam mid-credits scene?
Why does the Sun have different day lengths, but not the gas giants?
What are the purposes of autoencoders?
The screen of my macbook suddenly broken down how can I do to recover
Drawing ramified coverings with tikz
Is it possible to put a rectangle as background in the author section?
If a character has darkvision, can they see through an area of nonmagical darkness filled with lightly obscuring gas?
Which one is correct as adjective “protruding” or “protruded”?
How to bake one texture for one mesh with multiple textures blender 2.8
Should I outline or discovery write my stories?
Pre-mixing cryogenic fuels and using only one fuel tank
Freedom of speech and where it applies
Why did the Mercure fail?
why `nmap 192.168.1.97` returns less services than `nmap 127.0.0.1`?
GraphicsGrid with a Label for each Column and Row
What does dense mean in algebraic geometry?
Relation between varieties in the sense of Serre's FAC and algebraic schemesAffine algebraic sets are quasi-projective varietiesA doubt in the proof of Prop. 1.10 of Hartshorne's Algebraic GeometryDefinition of quasiprojective variety by ShafarevichZariski dense implies classically dense?A rational function on a dense subset of a varietyWhat does Liu mean by “topological open/closed immersion” in his book “Algebraic Geometry and Arithmetic Curves”?Regular functions on a varietyregular functions are determined only up to open setsQuestion on Hartshorne II.6.1
$begingroup$
If $f, g$ are regular functions on a variety X, then the set of points where $f-g=0$ is closed and dense, hence equal to $X. $ Why is it dense and why does the closure of such a set satisfy the equation?
An open subset of a variety is dense (which means that the closure of an open subset is the whole variety). Why is it dense? Many thanks for your comment.
algebraic-geometry
$endgroup$
add a comment |
$begingroup$
If $f, g$ are regular functions on a variety X, then the set of points where $f-g=0$ is closed and dense, hence equal to $X. $ Why is it dense and why does the closure of such a set satisfy the equation?
An open subset of a variety is dense (which means that the closure of an open subset is the whole variety). Why is it dense? Many thanks for your comment.
algebraic-geometry
$endgroup$
1
$begingroup$
To the latter question: this is because a variety is irreducible as a topological space. If $U$ is open, then $overline U$ and $Xsetminus U$ are closed and cover $X$, so either $U$ is empty or its closure is $X$. For the former, this is clearly not the case for arbitrary $f,g$. Are you perhaps working through a proof of the statement "if $f,g$ are equal on a dense set, they are equal everywhere"? If so, this set is dense by assumption.
$endgroup$
– Wojowu
Mar 13 at 21:55
2
$begingroup$
Your title and question are mismatched - are you actually interested in the meaning of dense in addition to the questions in your post? Secondly, you're missing important assumptions about $X$ and the locus where $f=g$ in your first question. If $g=f+1$, certainly they can't be equal anywhere.
$endgroup$
– KReiser
Mar 13 at 22:00
add a comment |
$begingroup$
If $f, g$ are regular functions on a variety X, then the set of points where $f-g=0$ is closed and dense, hence equal to $X. $ Why is it dense and why does the closure of such a set satisfy the equation?
An open subset of a variety is dense (which means that the closure of an open subset is the whole variety). Why is it dense? Many thanks for your comment.
algebraic-geometry
$endgroup$
If $f, g$ are regular functions on a variety X, then the set of points where $f-g=0$ is closed and dense, hence equal to $X. $ Why is it dense and why does the closure of such a set satisfy the equation?
An open subset of a variety is dense (which means that the closure of an open subset is the whole variety). Why is it dense? Many thanks for your comment.
algebraic-geometry
algebraic-geometry
edited Mar 13 at 23:29
Dzoooks
905416
905416
asked Mar 13 at 21:52
user249018user249018
435137
435137
1
$begingroup$
To the latter question: this is because a variety is irreducible as a topological space. If $U$ is open, then $overline U$ and $Xsetminus U$ are closed and cover $X$, so either $U$ is empty or its closure is $X$. For the former, this is clearly not the case for arbitrary $f,g$. Are you perhaps working through a proof of the statement "if $f,g$ are equal on a dense set, they are equal everywhere"? If so, this set is dense by assumption.
$endgroup$
– Wojowu
Mar 13 at 21:55
2
$begingroup$
Your title and question are mismatched - are you actually interested in the meaning of dense in addition to the questions in your post? Secondly, you're missing important assumptions about $X$ and the locus where $f=g$ in your first question. If $g=f+1$, certainly they can't be equal anywhere.
$endgroup$
– KReiser
Mar 13 at 22:00
add a comment |
1
$begingroup$
To the latter question: this is because a variety is irreducible as a topological space. If $U$ is open, then $overline U$ and $Xsetminus U$ are closed and cover $X$, so either $U$ is empty or its closure is $X$. For the former, this is clearly not the case for arbitrary $f,g$. Are you perhaps working through a proof of the statement "if $f,g$ are equal on a dense set, they are equal everywhere"? If so, this set is dense by assumption.
$endgroup$
– Wojowu
Mar 13 at 21:55
2
$begingroup$
Your title and question are mismatched - are you actually interested in the meaning of dense in addition to the questions in your post? Secondly, you're missing important assumptions about $X$ and the locus where $f=g$ in your first question. If $g=f+1$, certainly they can't be equal anywhere.
$endgroup$
– KReiser
Mar 13 at 22:00
1
1
$begingroup$
To the latter question: this is because a variety is irreducible as a topological space. If $U$ is open, then $overline U$ and $Xsetminus U$ are closed and cover $X$, so either $U$ is empty or its closure is $X$. For the former, this is clearly not the case for arbitrary $f,g$. Are you perhaps working through a proof of the statement "if $f,g$ are equal on a dense set, they are equal everywhere"? If so, this set is dense by assumption.
$endgroup$
– Wojowu
Mar 13 at 21:55
$begingroup$
To the latter question: this is because a variety is irreducible as a topological space. If $U$ is open, then $overline U$ and $Xsetminus U$ are closed and cover $X$, so either $U$ is empty or its closure is $X$. For the former, this is clearly not the case for arbitrary $f,g$. Are you perhaps working through a proof of the statement "if $f,g$ are equal on a dense set, they are equal everywhere"? If so, this set is dense by assumption.
$endgroup$
– Wojowu
Mar 13 at 21:55
2
2
$begingroup$
Your title and question are mismatched - are you actually interested in the meaning of dense in addition to the questions in your post? Secondly, you're missing important assumptions about $X$ and the locus where $f=g$ in your first question. If $g=f+1$, certainly they can't be equal anywhere.
$endgroup$
– KReiser
Mar 13 at 22:00
$begingroup$
Your title and question are mismatched - are you actually interested in the meaning of dense in addition to the questions in your post? Secondly, you're missing important assumptions about $X$ and the locus where $f=g$ in your first question. If $g=f+1$, certainly they can't be equal anywhere.
$endgroup$
– KReiser
Mar 13 at 22:00
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%2f3147236%2fwhat-does-dense-mean-in-algebraic-geometry%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%2f3147236%2fwhat-does-dense-mean-in-algebraic-geometry%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
1
$begingroup$
To the latter question: this is because a variety is irreducible as a topological space. If $U$ is open, then $overline U$ and $Xsetminus U$ are closed and cover $X$, so either $U$ is empty or its closure is $X$. For the former, this is clearly not the case for arbitrary $f,g$. Are you perhaps working through a proof of the statement "if $f,g$ are equal on a dense set, they are equal everywhere"? If so, this set is dense by assumption.
$endgroup$
– Wojowu
Mar 13 at 21:55
2
$begingroup$
Your title and question are mismatched - are you actually interested in the meaning of dense in addition to the questions in your post? Secondly, you're missing important assumptions about $X$ and the locus where $f=g$ in your first question. If $g=f+1$, certainly they can't be equal anywhere.
$endgroup$
– KReiser
Mar 13 at 22:00