Zac Maeder-Wolland: Some Daily AG Problems

Hartshorne Exercise II.7.3.

Let $\varphi: \PP{k}{n}\ra\PP{k}{m}$ be a morphism. Then:

(a) The image of $\varphi$ must be closed in $\PP{\C}{m}$ due to properness, so one case can certainly be that $\op{im}\varphi =\{\op{pt}\}$. Every morphism to $\PP{\C}{m}$ is determined by a linear system on the base, but the base is $\PP{\C}{n}$. Hence such a morphism amounts to the choice of an $(m+1)$-tuple of degree $d$ homogeneous polynomials $\{P_j\}_{j=0}^m$ in $n+1$ variables that that globally generate $\OO_{\PP{\C}{n}}(d)$. That is, $\mathcal O_{\PP{k}{n}}(d)\cong\varphi^*\OO_{\PP{k}{m}}(1)$. The choice of $m+1$ constants corresponds to the constant morphism with image a point. If, for a contradiction, $m<n$ and $\op{im}\varphi\ne \{\op{pt}\}$, then we’d have global sections $s_0,…,s_m$ that globally generate $\OO(d)$ with $d\ge 1$. Then the radical of the homogeneous ideal defined by these polynomials must be the irrelevant one (as ${s_i\ne 0}$ cover $\PP{k}{n}$), which contradicts Krull’s principle ideal theorem, i.e., in considering the affine cone, after adding all these equations we will have cut the dimension of $\A{k}{n+1}$ down to $0$, yet we would have only added $m+1<n+1$ equations, which is absurd.

To see why the image of non-constant $\varphi$ must have dimension $n$, suppose ab absurdo that $\dim\op{im}\varphi <n\le m$. Now, $\op{im}\varphi$ is a projective variety and is not all of $\PP{k}{m}$, so it has a nonzero homogeneous ideal, and is therefore contained in at least one hypersurface. Let $P$ be a point not on the hypersurface, and define a morphism $\op{im}\varphi\ra\PP{k}{m-1}$ as projection away from $P$, i.e. ($x\mapsto [L_1(x):…:L_m(x)]$ for $L_1,…,L_m$ linearly independent lines passing through $P$). If this morphism is surjective, then $\dim\op{im}\varphi=m-1$ and therefore $n=m$, hence we would obtain a non-constant morphism $\PP{k}{n}\ra\PP{k}{n-1}$ which contradicts our previous argument. If it is not surjective, then we continue in this fashion. At some point, we either obtain a non-constant morphism $\PP{k}{m-s}\ra\PP{k}{m-r-1}$ for some $0\le r\le m-n-1 $ and $0\le s\le r$ or, by composing all of them, a non-constant morphism $\PP{k}{n}\ra \PP{k}{n-1}$. In any case, we have a contradiction.

(b) The $d$ above is uniquely determined because $\mathcal O_{\PP{k}{n}}(d)\cong\varphi^*\OO_{\PP{k}{m}}(1)$ is unique. Let $\rho:\PP{k}{n}\ra\PP{k}{N}$ be the degree $d$ Veronese embedding, where $N={n+d\choose d}-1$ . Recall $P_j=\varphi^*x_j$ for $0\le j\le m$ are global sections of $\OO_{\PP{k}{n}}(d)$. The collection $M_0,…,M_{N}\in H^0(\PP{k}{n},\OO_{\PP{k}{n}}(d))$ of standard monomials in $n+1$ variables of degree $d$ form a basis. Consider the linear transformation $T\in M_{(m+1)\times (N+1)}$ that takes $M_0,…,M_{N}$ to ${P_j}_{j=0}^m$. Obviously it is surjective, and thus factors as

$$k^{N+1}\ra k^{N+1}/\op{Ker}T\xrightarrow{\cong}k^{m+1}.$$

As we saw in Problem 34 (Ex. II.7.2), this corresponds to a composition of morphisms,

$$\PP{k}{N}-L\xrightarrow{\pi} \PP{k}{m}\xrightarrow[\cong]{\sigma}\PP{k}{m}$$

where $L:=\Bbb P(\op{Ker}T)$ is a linear subspace of dimension $N-m-1$. Under this composition $\sigma\circ\pi\circ\rho$ , the sections $x_j\in H^0(\PP{k}{m},\OO_{\PP{k}{m}}(1))$ pull back to $P_j$. It follows that $\mathcal O_{\PP{k}{n}}(d)\cong\varphi^*\OO_{\PP{k}{m}}(1)\cong \rho^*\pi^*\sigma^* \OO_{\PP{k}{m}}(1)$ so $\varphi=\sigma\circ\pi\circ\rho$. Thus, we have given the desired factorization of $\varphi$. $\varphi$ has finite fibers because $\mathcal O_{\PP{k}{n}}(d)\cong\varphi^*\OO_{\PP{k}{m}}(1)$ is ample. In particular, if $\iota:F\ra \PP{k}{n}$ were any closed fiber of $\varphi$, then $\iota^*\OO(d)$ would be both ample and trivial on $F$, which is only possible of $\dim F=0$.