# How to define the Canonical Divisor on Weighted Projective Planes

Hi everybody!

Does anybody know some paper about how to define the canonical divisor on Weighted Projective Planes? I am also interested in an Adjunction Formula and an analogue of Riemann Roch Theorem for curves on this spaces. I have read a lot of it, there are a few papers related with the topic but I am interested in "down-to-earth" results.

Thank you very much!

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As weighted projective spaces are singular in general, one has to be careful when defining the canonical divisor. For a normal variety $X$ (like weighted projective space), one normally takes a canonical divisor on the smooth locus, and then takes its closure to be the definition of the canonical divisor on $X$. – Daniel Loughran Aug 2 '12 at 16:01
You can try the book of Fulton "Introduction to toric varieties" --- a weighted projective space is the simplest example of a toric variety. – Sasha Aug 3 '12 at 8:25

A nice characterization of the canonical divisor on a weighted projective space can be found in the book

Toric Varieties by David A. Cox, John B. Little and Hal Schenck, Graduate Studies in Mathematics, Volume 124, American Mathematical Society

They define the canonical divisor of a normal variety $X$ of dimension $k$ as the Weil divisor corresponding to the (canonical) sheaf of Zariski $k$-forms on $X$ (Definition 8.0.20; they prove all of this makes sense). A weighted projective spaces can be constructed as a toric variety from a fan (which implies it is a normal variety). Example 3.1.17 in the book gives the fan: Let $q_0,...,q_n \in \mathbb{N}$ satisfy $\rm{gcd}(q_0,...,q_n)=1$. Let $N= \mathbb{Z}^{n+1}/\mathbb{Z}\cdot (q_0,...,q_n)$ and let $u_0,...,u_n$ be the images in $N$ (by projection) of the standard basis vectors in $\mathbb{Z}^{n+1}$. The fan $\Sigma$ consists of all cones generated by all proper subsets of $\{u_0,...,u_n\}$. The orbit-cone correspondence (explained in Chapter 3) establishes the correspondence between the cones in $\Sigma$ and torus-invariant subvarieties of $X$; one-dimensional cones determine torus-invariant hypersurfaces. Theorem 8.2.3 (proved for general toric varieties coming from fans) says that the canonical divisor equals $-\sum_\rho D_\rho$, where $D_\rho$ is the torus-invariant (Weil) divisor corresponding to the cone $\rho$ and $\rho$ runs over all one-dimensional cones in the fan $\Sigma$. So in the case of a weighted projective space one gets the canonical divisor as $-\sum_{j=0}^n D_{\mathbb{R}_+u_j}$, just as suggested by the standard projective space.

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You can try the paper by Dolgachev Weighted Projective Varieties

In Section 3.3, page 54 there is the treatment of the dualizing sheaf of a weighted projective space or, more generally, of a quasi-smooth weighted projective intersection (see in particular Theorem 3.3.4, page 56)

Regarding Riemann-Roch, one needs to make some corrections to the usual formula, because of the presence of singularities. The exact expression of these corrections, in the case where the singularities are all isolated, can be found in M. Reid's paper "Young Person's Guide o Canonical Singularities", Algebraic Geometry, Bowdoin 1985, ed. S. Bloch, Proc. of Symposia in Pure Math. 46, A.M.S. (1987), vol. 1, 345--414. See in particular Chapter III "Contributions of $\mathbf{Q}$-divisors to $\mathbf{RR}$".

Unfortunately I could not find Reid's paper online. However, I'm quite sure that some electronic copies are circulating.

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See also C.Delorme: Espaces projectifs anisotropes (1975, Bull.S.M.F.103). – Al-Amrani Oct 23 '15 at 8:54
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