Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$, be the space of homogeneous degree $d$ polynomials in three variables $[X,Y,Z] \in \mathbb{P}^2$ upto scaling, where $\delta_d = \frac{d(d+3)}{2}$. Note that we have two tautological line bundles $$ \gamma_{\mathcal{D}} \rightarrow \mathcal{D}, \qquad \gamma_{\mathbb{P}^2} \rightarrow \mathbb{P}^2.$$
Note that we have a natural section of the line bundle $$ \psi: \mathcal{D} \times \mathbb{P}^2\rightarrow \gamma_{\mathcal{D}}^* \otimes \gamma_{\mathbb{P}^2}^{*d} $$ given by $$ \psi( [s], p) = s(p) $$ Let $$ g : \mathcal{D} \times \mathbb{P}^2 \rightarrow \mathcal{D} \times \mathbb{P}^2$$ be a diffeomorphism (need not be a biholomorphism).

Consider the subgroup under which $\psi$ is invariant, i.e. $$ \psi \circ g = \psi. $$ Does this subgroup act transitively on $\mathcal{D} \times \mathbb{P}^2$ ? In other words given any two points $([s_1], p_1)$ and $([s_2], [p_2])$ does there always exist a diffeomorphism preserving $\psi$ that takes one point to the other? More generally consider the section $$ \psi_2: \mathcal{D} \times ((\mathbb{P}^2)^2 - \Delta)\rightarrow \gamma_{\mathcal{D}}^* \otimes \gamma_{\mathbb{P}^2}^{*d} \oplus \gamma_{\mathcal{D}}^* \otimes \gamma_{\mathbb{P}^2}^{*d} $$ given by $$ \psi_2 ([s], p_1, p_2) = s(p_1), s(p_2).$$ Does the subgroup of $$\mathcal{Diff} ( \mathcal{D} \times ((\mathbb{P}^2)^2 - \Delta))$$ under which $\psi_2$ is invariant act transitively on $\mathcal{D} \times ((\mathbb{P}^2)^2 - \Delta)$? Here $\Delta$ is the diagonal. Finally I have the same question for $k$ distinct points.

share|improve this question

1 Answer 1

up vote 4 down vote accepted

I am not sure I understand the meaning of the equation $\psi\circ g=\psi$ as I don't see how to compare a section at two different points without fixing an isomorphism between the line-bundle and its pull-back under $g$. Anyway, in any possible interpretation, I don't think the group preserving $\psi$ acts transitively as it should preserve the zero locus of $\psi$.

share|improve this answer
    
You are right, my question makes no sense. I guess I meant to say that the section is $G$ equivariant in some sense, but for that there has to be an action of the group on the Vector bundle as well. Very naively what I was thinking is the following...... consider this section s([X,Y]) = X^2 + Y^2. Is there any sense in which I can say the section is ``invariant'' under the action of Z2 (i.e. [X,Y] going to [Y,X])? –  Ritwik Nov 2 '11 at 3:52

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.