Smoothness of Picard scheme of an abelian variety as in Mumford's book
up vote
2
down vote
favorite
In Mumford's book, he shows $Pic^0$ of an abelian scheme is smooth by considering obstruction class:
why is the homomorphism $mu^*-p_1^*-p_2^*$ is injective on the second chomology? Mumford said in the last paragraph that we shall use structure of cohomology of abelian varieties (it's a wedge product of first cohomology) and Kunneth formula, but I don't know how to deduce the injectivity from those facts.
algebraic-geometry abelian-varieties
add a comment |
up vote
2
down vote
favorite
In Mumford's book, he shows $Pic^0$ of an abelian scheme is smooth by considering obstruction class:
why is the homomorphism $mu^*-p_1^*-p_2^*$ is injective on the second chomology? Mumford said in the last paragraph that we shall use structure of cohomology of abelian varieties (it's a wedge product of first cohomology) and Kunneth formula, but I don't know how to deduce the injectivity from those facts.
algebraic-geometry abelian-varieties
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
In Mumford's book, he shows $Pic^0$ of an abelian scheme is smooth by considering obstruction class:
why is the homomorphism $mu^*-p_1^*-p_2^*$ is injective on the second chomology? Mumford said in the last paragraph that we shall use structure of cohomology of abelian varieties (it's a wedge product of first cohomology) and Kunneth formula, but I don't know how to deduce the injectivity from those facts.
algebraic-geometry abelian-varieties
In Mumford's book, he shows $Pic^0$ of an abelian scheme is smooth by considering obstruction class:
why is the homomorphism $mu^*-p_1^*-p_2^*$ is injective on the second chomology? Mumford said in the last paragraph that we shall use structure of cohomology of abelian varieties (it's a wedge product of first cohomology) and Kunneth formula, but I don't know how to deduce the injectivity from those facts.
algebraic-geometry abelian-varieties
algebraic-geometry abelian-varieties
asked Nov 23 at 2:14
zzy
2,2951419
2,2951419
add a comment |
add a comment |
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',
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%2f3009896%2fsmoothness-of-picard-scheme-of-an-abelian-variety-as-in-mumfords-book%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
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%2f3009896%2fsmoothness-of-picard-scheme-of-an-abelian-variety-as-in-mumfords-book%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