Cutting out a Riemann surface inside its Jacobian variety
After choosing a base point $P_{0}$ in a compact Riemann surface $X$ of genus $g$, the Abel-Jacobi map gives an embedding of $X$ into its Jacobian variety $Jac(X).$ This map can also be extended to define a function from the $m$-th symmetric power $S^{m}(X)$ to $Jac(X)$ which, according to Jacobi's inversion theorem, is surjective when $mgeq g.$
Riemann was able to identify the image of $S^{g-1}(X)$ in $Jac(X)$ as a translate of the zeroes of his theta function. Is there a similar description of the image of $S^{m}(X)$ for $1leq mleq g-1$? In particular, is it possible to cut out $X$ inside its Jacobian variety using some more or less canonically defined theta functions?
riemann-surfaces theta-functions
asked Dec 4 '18 at 16:36
Raul Gomez
1665
add a comment |
After choosing a base point $P_{0}$ in a compact Riemann surface $X$ of genus $g$, the Abel-Jacobi map gives an embedding of $X$ into its Jacobian variety $Jac(X).$ This map can also be extended to define a function from the $m$-th symmetric power $S^{m}(X)$ to $Jac(X)$ which, according to Jacobi's inversion theorem, is surjective when $mgeq g.$
Riemann was able to identify the image of $S^{g-1}(X)$ in $Jac(X)$ as a translate of the zeroes of his theta function. Is there a similar description of the image of $S^{m}(X)$ for $1leq mleq g-1$? In particular, is it possible to cut out $X$ inside its Jacobian variety using some more or less canonically defined theta functions?
riemann-surfaces theta-functions
asked Dec 4 '18 at 16:36
Raul Gomez
1665
add a comment |
After choosing a base point $P_{0}$ in a compact Riemann surface $X$ of genus $g$, the Abel-Jacobi map gives an embedding of $X$ into its Jacobian variety $Jac(X).$ This map can also be extended to define a function from the $m$-th symmetric power $S^{m}(X)$ to $Jac(X)$ which, according to Jacobi's inversion theorem, is surjective when $mgeq g.$
Riemann was able to identify the image of $S^{g-1}(X)$ in $Jac(X)$ as a translate of the zeroes of his theta function. Is there a similar description of the image of $S^{m}(X)$ for $1leq mleq g-1$? In particular, is it possible to cut out $X$ inside its Jacobian variety using some more or less canonically defined theta functions?
riemann-surfaces theta-functions
asked Dec 4 '18 at 16:36
Raul Gomez
1665
After choosing a base point $P_{0}$ in a compact Riemann surface $X$ of genus $g$, the Abel-Jacobi map gives an embedding of $X$ into its Jacobian variety $Jac(X).$ This map can also be extended to define a function from the $m$-th symmetric power $S^{m}(X)$ to $Jac(X)$ which, according to Jacobi's inversion theorem, is surjective when $mgeq g.$
Riemann was able to identify the image of $S^{g-1}(X)$ in $Jac(X)$ as a translate of the zeroes of his theta function. Is there a similar description of the image of $S^{m}(X)$ for $1leq mleq g-1$? In particular, is it possible to cut out $X$ inside its Jacobian variety using some more or less canonically defined theta functions?
riemann-surfaces theta-functions
riemann-surfaces theta-functions
asked Dec 4 '18 at 16:36
Raul Gomez
1665
asked Dec 4 '18 at 16:36
Raul Gomez
1665
asked Dec 4 '18 at 16:36
Raul Gomez
1665
asked Dec 4 '18 at 16:36
Raul Gomez
1665
asked Dec 4 '18 at 16:36
Raul Gomez
1665
1665
add a comment |
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%2f3025802%2fcutting-out-a-riemann-surface-inside-its-jacobian-variety%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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3025802%2fcutting-out-a-riemann-surface-inside-its-jacobian-variety%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
