# Schemes

\gdef\mc#1{\mathcal{#1}} \gdef\mf#1{\mathfrak{#1}}

# Projective Schemes

# The Proj Construction

Assume that RR is an integral graded ring, finitely-generated over R0=CR_{0} = \CC; such a ring R=C[x0,,xn]/IR = \CC[x_{0},\ldots, x_{n}]/I corresponds to an irreducible affine variety C(X)=Spec(R)=VA(I)An+1\mathcal{C}(X) = \Spec(R) = V_{\mathbb{A}}(I) \subseteq \mathbb{A}^{n+1}, where the subscript of VA(I)V_{\AA}(I) stands for the affine or inhomogeneous. C(X)\mc{C}(X) is called the (weighted homogeneous) affine cone over the projective variety VP(I)=XV_{\mathbb{P}}(I) = X, which is defined as the homogeneous spectrum:

X=Proj(R)=VP(I):=(VA(I){0})/GmP(a0,,an),X = \Proj(R) = V_{\mathbb{P}}(I) := (V_{\mathbb{A}}(I) - \{0\})/\mathbb{G}_{m} \subseteq \mathbb{P}(a_{0},\ldots,a_{n}),

where the a0,,anZ0a_{0},\ldots, a_{n} \in \mathbb{Z}_{\geq 0} are the weights of the Gm\GG_{m}-action on the coordinates x0,,xnx_{0},\ldots, x_{n} of C(X)\mathcal{C}(X).

::: remark Remark In the case when R0=CR_{0} = \CC (or any other field for that matter), Spec(R0)={0}\Spec(R_{0}) = \{0\} and Proj(R)\Proj(R) can be also viewed as Proj(R)=(Spec(R)Spec(R0))/Gm\Proj(R) = (\Spec(R) - \Spec(R_{0}))/\mathbb{G}_{m}. :::

::: remark Remark When RR is generated by R1R_{1} over R0R_{0}, then RR is a quotient of SymR0(R1)Sym_{R_{0}}(R_{1}), Spec(R)\Spec(R) is a subspace of Spec(R0)×R1\Spec(R_{0}) \times R_{1}^{\ast}, and Proj(R)\Proj(R) is a subspace of Spec(R0)×P(R1)\Spec(R_{0}) \times \mathbb{P}(R_{1}^{\ast}).

In particular, the inclusion of R0RR_{0} \hookrightarrow R induces a map from Proj(R)Spec(R0)\Proj(R) \rightarrow \Spec(R_{0}), whose fibres are subvarieties of P(R1)\mathbb{P}(R_{1}^{\ast}). For this reason, the variety Proj(R)\Proj(R) is often referred to as being projective over (an) affine. :::


Consider the polynomial ring R=C[x0,,xn]R = \CC[x_{0},\ldots, x_{n}] with the standard grading, so that the ideal II in the definition is trivial. Then C(X)=Spec(R)=VA({0})An+1\mathcal{C}(X) = \Spec(R) = V_{\mathbb{A}}(\{0\}) \cong \mathbb{A}^{n+1}. Then

X=Proj(R)=(Spec(R)Spec(R0)/C(An+1{0})/C=Pn,X = \Proj(R) = (\Spec(R) - \Spec(R_{0})/\CC^{\ast} \cong (\mathbb{A}^{n+1} - \{0\})/\CC^{\ast} = \mathbb{P}^{n},

as the affine cone of nn-dimensional projective space Pn\mathbb{P}^{n} is just (n+1)(n+1)-dimensional affine space An+1\mathbb{A}^{n+1}.

# Projective Scheme(s) over an Affine Scheme

(Hartshorne Corollary 5.16)

::: proposition Proposition Let RR be a ring.

  • If YY is a closed subscheme of PRn\PP_{R}^{n}, then there is a homogeneous ideal IS=R[x0,,xn]I \lhd S = R[x_{0},\ldots, x_{n}] such that YY is the closed subscheme determined by II.
  • A scheme YY over Spec(R)\Spec(R) is projective if and only if it is isomorphic to Proj(S)\Proj(S) for some graded ring SS, where S0=RS_{0} = R, and SS is finitely generated by S1S_{1} as an S0S_{0}-algebra.


# Truncated Rings and the Veronese Embedding

Let RR be a graded ring. The dd-th truncated ring is the subring R[d]RR^{[d]} \subset R defined by

R[d]:=dnRn=i1Rdi, R^{[d]} := \bigoplus_{d | n} R_{n} = \bigoplus_{i \geq 1}R_{di},

i.e we only consider the homogeneous pieces of RR that are of degree divisible by dd.


Let R=C[x0,x1]R = \CC[x_{0},x_{1}] which has Proj(R)=P1\Proj(R) = \mathbb{P}^{1}. Then R[2]=C[x02,x0x1,x12]C[a,b,c]/(acb2)R^{[2]} = \CC[x_{0}^{2},x_{0}x_{1}, x_{1}^{2}] \cong \CC[a,b,c]/(ac - b^{2}), and R[3]=C[x03,x02x1,x0x12,x13]C[a,b,c,d]/(adbc,acb2,bdc2)R^{[3]} = \CC[x_{0}^{3},x_{0}^{2}x_{1},x_{0}x_{1}^{2},x_{1}^{3}] \cong \CC[a,b,c,d]/(ad-bc, ac - b^{2}, bd - c^{2}), etc.

::: proposition Proposition For a graded ring RR as above,

Proj(R[d])Proj(R),for any d>0. \Proj(R^{[d]}) \cong \Proj(R),\qquad \text{for any } d> 0.


# References

[1] M. Reid, ‘Graded rings and varieties in weighted projective space’.

[2] N. Proudfoot, ‘Lectures on Toric Varieties’.