Timeline for Dual of a weighted projective space
Current License: CC BY-SA 3.0
6 events
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| Jan 22, 2013 at 23:38 | comment | added | Jason Starr | @Kimra: "What I would like to do is to find a one to one correspondence ..." What I am saying is that you are not likely to get such a correspondence, precisely because this relies on "bi-equivariance" rather than just "equivariance" in the classical case. | |
| Jan 18, 2013 at 20:59 | comment | added | Kimra | What I would like to do is to find a one to one correspondence between a point in the dual my wps and an surface in my original wps for example. Or more generally to be able to understand the dual of the wps through the original wps. Such a correspondence exists between point of the dual of the regular projective space and its hyperplanes. I am at the moment reading some theory of geometric invariant as I think that this should have led me to a dualise th wps , I don't know anything yet about toric varieties but I think that this is really interesting way to look on how to dualise a wps. | |
| Jan 18, 2013 at 17:44 | comment | added | Jason Starr |
For a finite dimensional vector space $V$, there is an isomorphism $\textbf{GL}_V \cong \textbf{GL}_{V^\vee}$, where $V^\vee$ is the dual vector space. Thus every (injective) morphism of group schemes $\lambda:\mathbb{G}_m\to \textbf{GL}_V$ with "positive weights" induces also $\lambda^\vee:\mathbb{G}_m \to \textbf{GL}_{V^\vee}$. However, the natural bilinear pairing $V\times V^\vee \to \mathbb{A}^1$ is "bi-equivariant" for $(\lambda,\lambda^vee)$ if and only if $\lambda$ is the center. So do not expect a universal family of Cartier divisors.
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| Jan 18, 2013 at 10:50 | comment | added | Dan Petersen | One way of understanding projective duality is through duality of toric varieties. Projective space is the toric variety of the standard $n$-simplex, which is a self-dual polytope. Weighted projective space is of course also toric which should give you a definition of its dual. I haven't worked out what the result is (hence why I'm posting this as a comment) but it shouldn't be hard, and I believe one should find that the dual of $\mathbf P(a_0,\ldots,a_n)$ is $\mathbf P(\frac d {a_0},\ldots, \frac d {a_n})$ where $d = \mathrm{lcm}(a_0,\ldots,a_n)$. | |
| Jan 18, 2013 at 9:36 | comment | added | Serge Lvovski | What properties of the "usual" projective duality do you want to generalize? | |
| Jan 17, 2013 at 21:21 | history | asked | Kimra | CC BY-SA 3.0 |