5
$\begingroup$

I have a fairly good understanding of what the dual of a projective space is. I am currently interested in weighted projective space but I haven't found anything on the construction of its dual space if it exists. At the same time I can't see how I can extend the definition of the usual dual projective space to the weighted one. Could anyone tell me if such a construction exists and if it does direct me to any usual reference of define it for me?

Thank you.

Improve this question
$\endgroup$
5
  • 1
    $\begingroup$ What properties of the "usual" projective duality do you want to generalize? $\endgroup$
    – Serge Lvovski
    Commented Jan 18, 2013 at 9:36
  • 7
    $\begingroup$ 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 $d = \mathrm{lcm}(a_0,\ldots,a_n)$. $\endgroup$
    – Dan Petersen
    Commented Jan 18, 2013 at 10:50
  • 1
    $\begingroup$ 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. $\endgroup$
    – Jason Starr
    Commented Jan 18, 2013 at 17:44
  • $\begingroup$ 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. $\endgroup$
    – Kimra
    Commented Jan 18, 2013 at 20:59
  • $\begingroup$ @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. $\endgroup$
    – Jason Starr
    Commented Jan 22, 2013 at 23:38

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .