Is $mathbb{A}^3 - {0}$ a separated variety?












1












$begingroup$


To check if $X=mathbb{A}^n -{0}$ is separated is equivalent to check that the diagonal is closed. In this case I think that: $X times X= {(x,y)| x,y in X }$ then $Delta(X)=Z(y-x)- {0}subset mathbb{A}^n $ but i'm not sure if this is closed or not. I think that it is open can you confirm that?






algebraic-geometry







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    To check if $X=mathbb{A}^n -{0}$ is separated is equivalent to check that the diagonal is closed. In this case I think that: $X times X= {(x,y)| x,y in X }$ then $Delta(X)=Z(y-x)- {0}subset mathbb{A}^n $ but i'm not sure if this is closed or not. I think that it is open can you confirm that?






    algebraic-geometry







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      To check if $X=mathbb{A}^n -{0}$ is separated is equivalent to check that the diagonal is closed. In this case I think that: $X times X= {(x,y)| x,y in X }$ then $Delta(X)=Z(y-x)- {0}subset mathbb{A}^n $ but i'm not sure if this is closed or not. I think that it is open can you confirm that?






      algebraic-geometry







      share|cite|improve this question









      $endgroup$




      To check if $X=mathbb{A}^n -{0}$ is separated is equivalent to check that the diagonal is closed. In this case I think that: $X times X= {(x,y)| x,y in X }$ then $Delta(X)=Z(y-x)- {0}subset mathbb{A}^n $ but i'm not sure if this is closed or not. I think that it is open can you confirm that?






      algebraic-geometry




      algebraic-geometry






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Feb 3 at 0:13









      andresandres

      2439




      2439






















          2 Answers
          2






          active

          oldest

          votes


















          2












          $begingroup$

          There are several errors in your description of $Delta(X)$. First, it lives inside $Xtimes X$, not $Bbb A^n$. Secondly, you're using $x$ and $y$ for points of $X$ and then also as coordinates (when you subtract them), which doesn't really make sense. The second error is maybe not so bad, but the first is serious: it gives you the wrong answer. This variety is separated, as the diagonal copy of $X$ is in fact closed inside $Xtimes X$, exactly as the zero locus of $x_i-y_i$ where the $x_i$ and $y_i$ are coordinates on the first and second $X$, respectively.






          share|cite|improve this answer









          $endgroup$





















            0












            $begingroup$

            The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes






            share|cite|improve this answer









            $endgroup$














              Your Answer








              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


















              StackExchange.ready(
              function () {
              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3097989%2fis-mathbba3-0-a-separated-variety%23new-answer', 'question_page');
              }
              );

              Post as a guest















              Required, but never shown

























              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              2












              $begingroup$

              There are several errors in your description of $Delta(X)$. First, it lives inside $Xtimes X$, not $Bbb A^n$. Secondly, you're using $x$ and $y$ for points of $X$ and then also as coordinates (when you subtract them), which doesn't really make sense. The second error is maybe not so bad, but the first is serious: it gives you the wrong answer. This variety is separated, as the diagonal copy of $X$ is in fact closed inside $Xtimes X$, exactly as the zero locus of $x_i-y_i$ where the $x_i$ and $y_i$ are coordinates on the first and second $X$, respectively.






              share|cite|improve this answer









              $endgroup$


















                2












                $begingroup$

                There are several errors in your description of $Delta(X)$. First, it lives inside $Xtimes X$, not $Bbb A^n$. Secondly, you're using $x$ and $y$ for points of $X$ and then also as coordinates (when you subtract them), which doesn't really make sense. The second error is maybe not so bad, but the first is serious: it gives you the wrong answer. This variety is separated, as the diagonal copy of $X$ is in fact closed inside $Xtimes X$, exactly as the zero locus of $x_i-y_i$ where the $x_i$ and $y_i$ are coordinates on the first and second $X$, respectively.






                share|cite|improve this answer









                $endgroup$
















                  2












                  2








                  2





                  $begingroup$

                  There are several errors in your description of $Delta(X)$. First, it lives inside $Xtimes X$, not $Bbb A^n$. Secondly, you're using $x$ and $y$ for points of $X$ and then also as coordinates (when you subtract them), which doesn't really make sense. The second error is maybe not so bad, but the first is serious: it gives you the wrong answer. This variety is separated, as the diagonal copy of $X$ is in fact closed inside $Xtimes X$, exactly as the zero locus of $x_i-y_i$ where the $x_i$ and $y_i$ are coordinates on the first and second $X$, respectively.






                  share|cite|improve this answer









                  $endgroup$



                  There are several errors in your description of $Delta(X)$. First, it lives inside $Xtimes X$, not $Bbb A^n$. Secondly, you're using $x$ and $y$ for points of $X$ and then also as coordinates (when you subtract them), which doesn't really make sense. The second error is maybe not so bad, but the first is serious: it gives you the wrong answer. This variety is separated, as the diagonal copy of $X$ is in fact closed inside $Xtimes X$, exactly as the zero locus of $x_i-y_i$ where the $x_i$ and $y_i$ are coordinates on the first and second $X$, respectively.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Feb 3 at 0:34









                  KReiserKReiser

                  10.1k21435




                  10.1k21435























                      0












                      $begingroup$

                      The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes






                      share|cite|improve this answer









                      $endgroup$


















                        0












                        $begingroup$

                        The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes






                        share|cite|improve this answer









                        $endgroup$
















                          0












                          0








                          0





                          $begingroup$

                          The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes






                          share|cite|improve this answer









                          $endgroup$



                          The answer to the title question is yes: a locally closed subvariety of a separated variety is separated. Cf. these notes







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Feb 3 at 0:39









                          BenBen

                          4,373617




                          4,373617






























                              draft saved

                              draft discarded




















































                              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.




                              draft saved


                              draft discarded














                              StackExchange.ready(
                              function () {
                              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3097989%2fis-mathbba3-0-a-separated-variety%23new-answer', 'question_page');
                              }
                              );

                              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







                              Popular posts from this blog

                              Can a sorcerer learn a 5th-level spell early by creating spell slots using the Font of Magic feature?

                              Image
                              28 6 $begingroup$ Per the Font of Magic feature, sorcerer can use Flexible Casting to create 5th-level spell slots at level 7, even though they are not typically available until level 9. You can transform unexpended sorcery points into one spell slot as a bonus action on your turn. The Creating Spell Slots table shows the cost of creating a spell slot of [5th level is 7] If such a sorcerer levels up while still having this 5th-level spell slot, can they choose a 5th-level spell as the spell they gain upon levelling up? The Spellcasting feature states: Additionally, when you gain a level in this class, you can choose one of the sorcerer spells you know and replace it with another spell from the sorcerer spell list, which also must be of a level for which you have spell slots.
                              Read more

                              Does disintegrating a polymorphed enemy still kill it after the 2018 errata?

                              Image
                              up vote 13 down vote favorite 1 After the 2018 errata, the disintegrate spell description now reads: A creature targeted by this spell must make a Dexterity saving throw. On a failed save, the target takes 10d6 + 40 force damage. The target is disintegrated if this damage leaves it with 0 hit points. If you were to polymorph an enemy into a rat and then disintegrate it, would the enemy be disintegrated or would it just return to its original form? I know that for Druids, it’s not an instant kill anymore, but is this the case for polymorph as well? dnd-5e spells polymorph errata share | improve this question edited 18 hours ago
                              Read more

                              A Topological Invariant for $pi_3(U(n))$

                              Image
                              3 3 $begingroup$ Recently, I saw a construction of topological invariant for $pi_3(U(n))$ with $ngeq 2$ : $$ N=frac{1}{24pi^2}int_{S^3} d^3x epsilon^{ijk} Tr[(U^{-1}partial_{x_i}U)(U^{-1}partial_{x_j}U)(U^{-1}partial_{x_k}U)] , $$ where $Uin U(n)$ depends on $boldsymbol{x}=(x_1,x_2,x_3)in S^3$ , $epsilon^{ijk}$ is the Levi-Civita symbol, $i,j,k=1,2,3$ , and the duplicated indexes are summed over. It is claimed that $N$ is an integer, but why? Update 02/02/2019 I think I got an argument for $n=2$ . In this case, $U=e^{i varphi} q$ with $qin SU(2)$ . Due to the trace and the Levi-Civita symbol in $N$ , $varphi$ does not contribute to $N$ . As $Tr[q^{dagger}partial_i q]=0$ and $(q^{dagger}partial_i q)^{dagger}=-q^{dagger}partial_i q$ , $q^{dagger}partial_i q$ in geneeral has the form $q^{dagger}partia
                              Read more