Let \(X\) be a scheme over a field \(k\). Let \(\mcl\) be an invertible sheaf on \(X\), and let \(\ts{s_0, \ldots, s_n}\) and \(\ts{t_0, \ldots, t_m}\) be two sets of sections of \(\mcl\) which generate the same subspace \(V \subseteq \Gamma(X, \mcl)\), and which generate the sheaf \(\mcl\) at every point. Suppose \(n \leq m\).
Show that the corresponding morphisms \(\varphi: X \to \PP^n_k\) and \(\psi: X \to \PP^m_k\) differ by a suitable linear projection \(\PP^m - L \to \PP^n\) and an automorphism of \(\PP^n\), where \(L\) is a linear subspace of \(\PP^m\) of dimension \(m - n - 1\).
Solution
Let \(E=k^{n+1}\) and \(F=k^{m+1}\), with standard bases \(e_i\) and \(f_j\). The two lists give surjective linear maps \[ S:E\longrightarrow V,\quad e_i\longmapsto s_i, \qquad T:F\longrightarrow V,\quad f_j\longmapsto t_j. \] Write \(r=\dim_kV\).
1. There is an injective linear map \(J:E\to F\) with \(T\circ J=S\).
Choose complements \(E=E_0\oplus\ker S\) and \(F=F_0\oplus\ker T\). The restrictions \(S|_{E_0}\) and \(T|_{F_0}\) are isomorphisms onto \(V\). Since \[ \dim\ker S=n+1-r\le m+1-r=\dim\ker T, \] choose an injection \(J_1:\ker S\to\ker T\). Define \(J\) on \(E_0\) to be \((T|_{F_0})^{-1}\circ S|_{E_0}\), and on \(\ker S\) to be \(J_1\). The images of these two summands lie in the complementary subspaces \(F_0\) and \(\ker T\), and both restrictions are injective. Thus \(J\) is injective, and its construction gives \(T\circ J=S\).
2. There is a matrix \(A=(a_{ij})\) of row rank \(n+1\) such that \(s_i=\sum_{j=0}^m a_{ij}t_j\).Its linear forms define a projection \(p_A:\PP^m\setminus L\to\PP^n\), with \(\dim L=m-n-1\) when \(m>n\).
Write \(J(e_i)=\sum_j a_{ij}f_j\). The identity in step 1 gives the required section identities, and injectivity of \(J\) makes the rows of \(A\) linearly independent. For homogeneous coordinates \(z_0,\ldots,z_m\), put \[ \ell_i(z)=\sum_j a_{ij}z_j,\qquad L=V_+(\ell_0,\ldots,\ell_n),\qquad p_A([z])=[\ell_0(z):\cdots:\ell_n(z)]. \] The kernel of the surjection \(A:k^{m+1}\to k^{n+1}\) has dimension \(m-n\), so its projectivization is the stated center. Choose a vector-space complement \(W\) to this kernel. Projection along the kernel maps \(\PP^m\setminus L\) to the linear \(n\)-plane of lines in \(W\), and \(A|_W\) identifies that plane with \(\PP^n\). Thus \(p_A\) is a linear projection followed by a projective linear isomorphism of its target. Under any fixed linear identification of that target with \(\PP^n\), the latter is an automorphism of \(\PP^n\). When \(m=n\), \(A\) is invertible, the center is empty, and \(p_A\) itself is a projective linear automorphism; the dimension \(-1\) convention refers to this empty center.
3. The image of \(\psi\) avoids \(L\), and \(\varphi=p_A\circ\psi\) as morphisms of schemes.
Under the isomorphism \(\psi^*\OO_{\PP^m}(1)\cong\mcl\), the pullback of \(\ell_i\) is \(\sum_j a_{ij}t_j=s_i\) [@Har10a, Theorem II.7.1]. The sections \(s_i\) generate \(\mcl\) at every point, so their common zero subscheme is empty. Consequently \(\psi^{-1}(L)=\varnothing\), and the composite is defined on all of \(X\).
Let \(X_{s_i}\) be the open subset where \(s_i\) generates \(\mcl\). On this open set, the coordinate ratio \(w_h/w_i\) on the \(i\)th affine chart of \(\PP^n\) pulls back along the composite to \[ \frac{\sum_j a_{hj}t_j}{\sum_j a_{ij}t_j}=\frac{s_h}{s_i}. \] These are exactly the coordinate-ring maps defining \(\varphi\) on \(X_{s_i}\). The opens \(X_{s_i}\) cover \(X\), so the morphisms agree, including when \(X\) is nonreduced. For \(X=\varnothing\) the same equality is the uniqueness of its morphism to the target.
4. Q.E.D.
Steps <1>1–<1>2 construct a projection with the required center without assuming either generating list is a basis. Step 3 proves the required factorization, and the target identification in step 2 gives its stated automorphism.