User tonywang - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T22:30:21Z http://mathoverflow.net/feeds/user/19905 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74214/examples-where-its-useful-to-know-that-a-mathematical-object-belongs-to-some-fam/83531#83531 Answer by tonywang for Examples where it's useful to know that a mathematical object belongs to some family of objects tonywang 2011-12-15T16:50:16Z 2011-12-15T16:50:16Z <p>There are $12$ elliptic plane cubics with fixed $j$ invariant passing through $8$ points in the plane ($4$ or $6$ if $j = 0, 1728$). </p> <p>Proof: Use the pencil of cubics argument to show that there are $12$ nodal cubis pasing through $8$ points. Then move the result along the family $\mathcal M_{1,1}$.</p> http://mathoverflow.net/questions/83392/h1-of-the-pull-back-of-the-tangent-bundle $H^1$ of the pull back of the tangent bundle. tonywang 2011-12-14T04:22:54Z 2011-12-14T20:05:19Z <p>If $C$ is a smooth elliptic curve and $f: C \to \mathbb P^n$, then $H^1(C,f^*T_{\mathbb P^n}) = 0.$ How do I prove this? The implication is that map from $C$ to $\mathbb P^n$ is unobstructed.</p> http://mathoverflow.net/questions/83270/lifting-of-torus-action-to-line-bundle Lifting of torus action to line bundle tonywang 2011-12-12T19:18:27Z 2011-12-12T19:18:27Z <p>Let $\mathbb{P}(V) = \mathbb{P}(\mathbb C \oplus \mathbb C)$ be with a $\mathbb C^*$ action : $\lambda (u,v) = (u,\lambda v)$. There are two fixed points of this action, say $0$ and $\infty$. What does this mean to equivariantly lift this action to a line bundle $L$ over $\mathbb P(V)$, by choosing the weights $[l_0, l_{\infty}]$ of the fibre representations $L_0, L_{\infty}$ at the fixed points?</p>