# Embed a weighted projective space into an unweighted projective space.

To show is the following.

Let $$X = P(a_0,\dotsc,a_n)$$, $$a_i \geq 1$$ be a weighted projective space (that is $$X = \operatorname{Proj} k[x_0,\dotsc,x_n]$$, where $$\operatorname{deg} x_i = a_i$$). Then there exists an $$N$$ and a closed embedding $$X \to \mathbb P^N_k$$ ($$\mathbb P^N_k = \operatorname{Proj} k[y_0,\dotsc,y_N]$$, where $$\operatorname{deg}y_i = 1$$).

I can't find a surjective ring morphism between the graded rings (there doesn't really exist one, right?). Is the embedding some kind of Veronese-like map?

There is a very ampleness condition due to Delorme in [Del75a]. You can also look at [BR86] for an English reference. We have changed the dimension $$n$$ in your notation to an $$r$$ to closer match the notation in these references.

Definition [Del75a, Def. 2.1; BR86, Defs. 4B.1, 4B.2, and 4B.3]. We let $$m_J := \operatorname{lcm}\{a_i \mid i \in J\}$$ for every non-empty subset $$J \subseteq \{0,1,\ldots,r\}$$, and let $$m := m_{\{1,2,\ldots,r\}}$$. Now set $$G := \begin{cases} -a_r & \text{if r = 0, and}\\ \displaystyle-\sum_{i=0}^r a_i + \frac{1}{r}\sum_{2 \le \nu \le r+1} \binom{r-1}{\nu-2}^{-1} \sum_{\lvert J \rvert = \nu} m_J & \text{otherwise.} \end{cases}$$ We then say that an integer $$n \ge 0$$ satisfies condition $$D(n)$$ if one of the following equivalent conditions hold:

• Given a relation $$\sum_{i = 0}^r B_ia_i = n + km$$ with $$k \in \mathbf{Z}_{>0}$$ and $$B_i \in \mathbf{Z}_{\ge0}$$ for every $$i$$, there exist $$b_i \in \mathbf{Z}_{\ge0}$$ with $$B_i \ge b_i$$ for every $$i$$, such that $$\sum_{i = 0}^r b_ia_i=km$$.

• Every monomial $$\prod_{i=0}^r x_i^{B_i}$$ of degree $$n+km$$ is divisible by a monomial $$\prod_{i=0}^r x_i^{b_i}$$ of degree $$km$$.

We then define $$F$$ to be the smallest integer such that $$n > F$$ implies $$D(n)$$ holds. We also define $$E$$ to be the smallest integer such that $$n > E$$ implies $$D(mn)$$ holds. Note that $$mE \le F$$.

One can then show that $$F$$ is finite and $$F \le G$$ [Del75a, Prop. 2.2; BR86, Prop. 4B.5]. The proof is a double induction on $$k$$ and $$r$$, and boils down to the pigeon-hole principle.

With notation as above, we then have the following:

Theorem [Del75a, Prop. 2.3; BR86, Thm. 4B.7]. Let $$X = \mathbf{P}(a_0,a_1,\ldots,a_r)$$ be a weighted projective space over a commutative ring $$A$$, where $$a_i \ge 1$$ for every $$1$$. With notation as above, we have the following:

1. $$\mathcal{O}_X(m)$$ is an ample invertible sheaf.

2. If $$n > F$$, then the sheaf $$\mathcal{O}_X(n)$$ is globally generated.

3. If $$n > 0$$ and $$n > E$$, then the sheaf $$\mathcal{O}_X(nm)$$ is very ample.

We prove (3), since this is what you are interested in. For every $$p \in \mathbf{Z}_{>0}$$, the condition $$D(mn)$$ implies that every monomial of degree $$pmn = (p-1)mn + mn$$ in $$A[x_0,x_1,\ldots,x_r]$$ is divisible by a monomial of degree $$(p-1)mn$$. Thus, the $$mn$$th Veronese subring $$A[x_0,x_1,\ldots,x_r]^{(mn)}$$ of $$A[x_0,x_1,\ldots,x_r]$$ is generated in degree $$1$$ over $$A$$.

You can therefore embed $$X$$ into $$\mathbf{P}^N_A$$, where $$N$$ is the number of generators of $$A[x_0,x_1,\ldots,x_r]_{mn}$$, by $$X = \operatorname{Proj}\bigl(A[x_0,x_1,\ldots,x_r]\bigr) \simeq \operatorname{Proj}\bigl(A[x_0,x_1,\ldots,x_r]^{(mn)}\bigr) \hookrightarrow \mathbf{P}^N_A.$$ Unraveling the definitions given above, we note that in particular, setting $$n = \biggl\lfloor\frac{1}{m}G\biggr\rfloor+1$$ works.

## References

[BR86] Mauro Beltrametti and Lorenzo Robbiano, "Introduction to the theory of weighted projective spaces," Exposition. Math. 4 (1986), no. 2, 111–162. mr: 879909.

[Del75a] Charles Delorme, "Espaces projectifs anisotropes," Bull. Soc. Math. France 103 (1975), no. 2, 203–223. doi: 10.24033/bsmf.1802. mr: 404277.

[Del75b] Charles Delorme, "Erratum: 'Espaces projectifs anisotropes'," Bull. Soc. Math. France 103 (1975), no. 4, 510. doi: 10.24033/bsmf.1812. mr: 404278.