User john - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T03:21:15Z http://mathoverflow.net/feeds/user/13975 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64089/general-point-general-line general point, general line John 2011-05-06T06:14:45Z 2011-05-09T22:19:52Z <p>Hello, could anyone explain the notion of ''general point'' and ''general line'', ''general hyperplane'' in algebraic geometry, What does it mean exactly general line in the 3 dimensional projective space? Thank you.</p> http://mathoverflow.net/questions/59898/algebraic-varieties-isomorphism-normal-reduced Algebraic varieties , isomorphism, normal, reduced.... John 2011-03-28T21:41:26Z 2011-03-28T21:41:26Z <p>Hello, could someboty help me, every advice or hint will be very helpful (or if you can tell me some good book). Thank you so much!</p> <ol> <li><p>X and Y are two varieties and there are points P of X et Q of Y such that the local rings O(X,P) and O(Y,Q) are isomorphic as C-algebras. I should show that there are open sets U in X ( P is in U) and V in Y ( Q is in V) and an isomorphism of U to V that sends P to Q. How can I construct that isomorphism? Is this true for projective varieties or only for affine?</p></li> <li><ul> <li>If Y is a complex algebraic variety and Z an open subvariety of Y. How can I prove that C(Z) = C(Y)</li> <li>The ring of regular functions on A² \ V(x) is isomorphic to C[x,y][1/x] ( where A² = Spec C[x,y] and C[x,y][1/x] is the localisation of C[x,y] in the multiplicative set {1,x,x^2,,,,}</li> <li>The ring of regular functions on A² \ {0} is isomorphic to C[x,y]</li> <li>How to conclude that A² \ {0} is neither affine or projective </li> </ul></li> <li><p>If X is a connected algebraic variety, I should prove that X is reduced ( resp. integral, resp. normal) if and only if, for every closed point p of X the local ring O(X,p) is reduced ( resp. integral, normal).</p></li> </ol>