Fdtxjr

To show is the following.

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

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.

We start with some definitions.

Definition [Del75a, Def. 2.1; BR86, Defs. 4B.1, 4B.2, and 4B.3]. We let $m_J := operatornamelcma_i mid i in J$ for every non-empty subset $J subseteq 0,1,ldots,r$, and let $m := m_1,2,ldots,r$. Now set
$$G := begincases
-a_r & textif $r = 0$, and\
displaystyle-sum_i=0^r a_i + frac1rsum_2 le nu le r+1 binomr-1nu-2^-1 sum_lvert J rvert = nu m_J & textotherwise.
endcases$$
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 $k in mathbfZ_>0$ and $B_i in mathbfZ_ge0$ for every $i$, there exist $b_i in mathbfZ_ge0$ 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 $X = mathbfP(a_0,a_1,ldots,a_r)$ 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. $mathcalO_X(m)$ is an ample invertible sheaf.




2. If $n > F$, then the sheaf $mathcalO_X(n)$ is globally generated.




3. If $n > 0$ and $n > E$, then the sheaf $mathcalO_X(nm)$ is very ample.

We prove (3), since this is what you are interested in. For every $p in mathbfZ_>0$, 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 $mathbfP^N_A$, where $N$ is the number of generators of $A[x_0,x_1,ldots,x_r]_mn$, by
$$X = operatornameProjbigl(A[x_0,x_1,ldots,x_r]bigr) simeq operatornameProjbigl(A[x_0,x_1,ldots,x_r]^(mn)bigr) hookrightarrow mathbfP^N_A.$$
Unraveling the definitions given above, we note that in particular, setting
$$n = biggllfloorfrac1mGbiggrrfloor+1$$
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.