After a six-week moratorium, we are back. I plan to keep posting problems and solutions daily, although this plan may be overly ambitious this quarter due to grad school applications. Regardless, certainly more to come (even if it might turn to semi-daily).
Hartshorne Exercise II.7.2.
Let $X$ be a scheme over a field $k$. Let $\mathcal{L}$ be an invertible sheaf on $X$, and let ${s_0, … ,s_n}$ and ${t_0, … ,t_m}$ be two sets of sections of $\mathcal L$, which generate the same subspace $V\subseteq H^0(X,\mathcal L)$, and which generate the sheaf $\mathcal L$ at every point. Suppose $n\le m$. Show that the corresponding morphisms $\varphi: X\ra \PP{\C}{n}$ and $\psi: X\ra \PP{\C}{m}$ differ by a suitable linear projection $\PP{\C}{m}-L\longrightarrow\PP{\C}{n}$ and an automorphism of $\PP{\C}{n}$, where $L$ is a linear subspace of $\PP{\C}{n}$ of dimension $m-n-1$.
Let $\dim V=r+1\le n+1$. We may assume that $\beta={t_0,…,t_r}$ and $\beta ‘={s_0,…,s_r}$ form bases for $V$. Let $P\in M_{r+1}(k)$ be the automorphism of $V$ that takes $\beta$ to $\beta ‘$, and for each $r+1\le i\le n$ write $s_i=\sum_{j=0}^ra_{ij}t_j$ with constants $a_{ij}\in k$. Let $T\in M_{(n+1)\times (m+1)}(k) $ be the block matrix whose upper left block is $P$, and whose lower left block is the $(n-r)\times (r+1)$ matrix with rows given as the $\boldsymbol{a_i}$ from above. The rest of the entries are $0$. Then $T$ takes the vector $(t_0,…,t_m)$ to $(s_0,…,s_n)$. The kernel of $T$ has dimension $(m+1)-(r+1)=m-r$ because its rank is $r+1$. Then $T$ determines the projection,
$$\pi:\PP{\C}{m}-\Bbb P(\op{Ker}T)\longrightarrow \PP{\C}{n}$$
that composes with $\psi$ to give $\varphi$. Here, $\Bbb P(\op{Ker}T)$ is the projective linear subspace corresponding to $\op{Ker}(T)$, and thus has dimension $m-r-1$. In cash, $\op{Ker}T\subset H^0(\PP{\C}{m},\OO (1))$ is the subspace spanned by $\big\{\sum_{j=0}^ra_{ij}X_j\big \}$, where the $X_j$ are projective coordinates. Note that $T$ factors as $k^{m+1}\ra k^{m+1}/\op{Ker}T\xrightarrow{\cong}k^{n+1}$, which corresponds to the desired composition of morphisms (the last isomorphism corresponds to an automorphism in $\op{PGL}(n,k)$ and thus of $\PP{\C}{n}$).
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