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.


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 $\psi$.

  • $\begingroup$ 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])? $\endgroup$ – Ritwik Nov 2 '11 at 3:52

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