Lemma 43.23.1. Let $V$ be a vector space of dimension $n + 1$. Let $X \subset \mathbf{P}(V)$ be a closed subscheme. If $X \not= \mathbf{P}(V)$, then there is a nonempty Zariski open $U \subset \mathbf{P}(V)$ such that for all closed points $p \in U$ the restriction of the projection $r_ p$ defines a finite morphism $r_ p|_ X : X \to \mathbf{P}(W_ p)$.
43.23 Projections
Recall that we are working over a fixed algebraically closed ground field $\mathbf{C}$. If $V$ is a finite dimensional vector space over $\mathbf{C}$ then we set
\[ \mathbf{P}(V) = \text{Proj}(\text{Sym}(V)) \]
where $\text{Sym}(V)$ is the symmetric algebra on $V$ over $\mathbf{C}$. See Constructions, Example 27.21.2. The normalization is chosen such that $V = \Gamma (\mathbf{P}(V), \mathcal{O}_{\mathbf{P}(V)}(1))$. Of course we have $\mathbf{P}(V) \cong \mathbf{P}^ n_{\mathbf{C}}$ if $\dim (V) = n + 1$. We note that $\mathbf{P}(V)$ is a nonsingular projective variety.
Let $p \in \mathbf{P}(V)$ be a closed point. The point $p$ corresponds to a surjection $V \to L_ p$ of vector spaces where $\dim (L_ p) = 1$, see Constructions, Lemma 27.12.3. Let us denote $W_ p = \mathop{\mathrm{Ker}}(V \to L_ p)$. Projection from $p$ is the morphism
\[ r_ p : \mathbf{P}(V) \setminus \{ p\} \longrightarrow \mathbf{P}(W_ p) \]
of Constructions, Lemma 27.11.1. Here is a lemma to warm up.
Proof. We claim the lemma holds with $U = \mathbf{P}(V) \setminus X$. For a closed point $p$ of $U$ we indeed obtain a morphism $r_ p|_ X : X \to \mathbf{P}(W_ p)$. This morphism is proper because $X$ is a proper scheme (Morphisms, Lemmas 29.43.5 and 29.41.7). On the other hand, the fibres of $r_ p$ are affine lines as can be seen by a direct calculation. Hence the fibres of $r_ p|X$ are proper and affine, whence finite (Morphisms, Lemma 29.44.11). Finally, a proper morphism with finite fibres is finite (Cohomology of Schemes, Lemma 30.21.1). $\square$
Lemma 43.23.2. Let $V$ be a vector space of dimension $n + 1$. Let $X \subset \mathbf{P}(V)$ be a closed subvariety. Let $x \in X$ be a nonsingular point.
If $\dim (X) < n - 1$, then there is a nonempty Zariski open $U \subset \mathbf{P}(V)$ such that for all closed points $p \in U$ the morphism $r_ p|_ X : X \to r_ p(X)$ is an isomorphism over an open neighbourhood of $r_ p(x)$.
If $\dim (X) = n - 1$, then there is a nonempty Zariski open $U \subset \mathbf{P}(V)$ such that for all closed points $p \in U$ the morphism $r_ p|_ X : X \to \mathbf{P}(W_ p)$ is étale at $x$.
Proof. Proof of (1). Note that if $x, y \in X$ have the same image under $r_ p$ then $p$ is on the line $\overline{xy}$. Consider the finite type scheme
\[ T = \{ (y, p) \mid y \in X \setminus \{ x\} ,\ p \in \mathbf{P}(V),\ p \in \overline{xy}\} \]
and the morphisms $T \to X$ and $T \to \mathbf{P}(V)$ given by $(y, p) \mapsto y$ and $(y, p) \mapsto p$. Since each fibre of $T \to X$ is a line, we see that the dimension of $T$ is $\dim (X) + 1 < \dim (\mathbf{P}(V))$. Hence $T \to \mathbf{P}(V)$ is not surjective. On the other hand, consider the finite type scheme
\[ T' = \{ p \mid p \in \mathbf{P}(V) \setminus \{ x\} , \ \overline{xp}\text{ tangent to }X\text{ at }x\} \]
Then the dimension of $T'$ is $\dim (X) < \dim (\mathbf{P}(V))$. Thus the morphism $T' \to \mathbf{P}(V)$ is not surjective either. Let $U \subset \mathbf{P}(V) \setminus X$ be nonempty open and disjoint from these images; such a $U$ exists because the images of $T$ and $T'$ in $\mathbf{P}(V)$ are constructible by Morphisms, Lemma 29.22.2. Then for $p \in U$ closed the projection $r_ p|_ X : X \to \mathbf{P}(W_ p)$ is injective on the tangent space at $x$ and $r_ p^{-1}(\{ r_ p(x)\} ) = \{ x\} $. This means that $r_ p$ is unramified at $x$ (Varieties, Lemma 33.16.8), finite by Lemma 43.23.1, and $r_ p^{-1}(\{ r_ p(x)\} ) = \{ x\} $ thus Étale Morphisms, Lemma 41.7.3 applies and there is an open neighbourhood $R$ of $r_ p(x)$ in $\mathbf{P}(W_ p)$ such that $(r_ p|_ X)^{-1}(R) \to R$ is a closed immersion which proves (1).
Proof of (2). In this case we still conclude that the morphism $T' \to \mathbf{P}(V)$ is not surjective. Arguing as above we conclude that for $U \subset \mathbf{P}(V)$ avoiding $X$ and the image of $T'$, the projection $r_ p|_ X : X \to \mathbf{P}(W_ p)$ is étale at $x$ and finite. $\square$
Lemma 43.23.3. Let $V$ be a vector space of dimension $n + 1$. Let $Y, Z \subset \mathbf{P}(V)$ be closed subvarieties. There is a nonempty Zariski open $U \subset \mathbf{P}(V)$ such that for all closed points $p \in U$ we have with $E \subset Y$ closed and $\dim (E) \leq \dim (Y) + \dim (Z) + 1 - n$.
\[ Y \cap r_ p^{-1}(r_ p(Z)) = (Y \cap Z) \cup E \]
Proof. Set $Y' = Y \setminus Y \cap Z$. Let $y \in Y'$, $z \in Z$ be closed points with $r_ p(y) = r_ p(z)$. Then $p$ is on the line $\overline{yz}$ passing through $y$ and $z$. Consider the finite type scheme
\[ T = \{ (y, z, p) \mid y \in Y', z \in Z, p \in \overline{yz}\} \]
and the morphism $T \to \mathbf{P}(V)$ given by $(y, z, p) \mapsto p$. Observe that $T$ is irreducible and that $\dim (T) = \dim (Y) + \dim (Z) + 1$. Hence the general fibre of $T \to \mathbf{P}(V)$ has dimension at most $\dim (Y) + \dim (Z) + 1 - n$, more precisely, there exists a nonempty open $U \subset \mathbf{P}(V) \setminus (Y \cup Z)$ over which the fibre has dimension at most $\dim (Y) + \dim (Z) + 1 - n$ (Varieties, Lemma 33.20.4). Let $p \in U$ be a closed point and let $F \subset T$ be the fibre of $T \to \mathbf{P}(V)$ over $p$. Then
\[ (Y \cap r_ p^{-1}(r_ p(Z))) \setminus (Y \cap Z) \]
is the image of $F \to Y$, $(y, z, p) \mapsto y$. Again by Varieties, Lemma 33.20.4 the closure of the image of $F \to Y$ has dimension at most $\dim (Y) + \dim (Z) + 1 - n$. $\square$
Lemma 43.23.4. Let $V$ be a vector space. Let $B \subset \mathbf{P}(V)$ be a closed subvariety of codimension $\geq 2$. Let $p \in \mathbf{P}(V)$ be a closed point, $p \not\in B$. Then there exists a line $\ell \subset \mathbf{P}(V)$ with $\ell \cap B = \emptyset $. Moreover, these lines sweep out an open subset of $\mathbf{P}(V)$.
Proof. Consider the image of $B$ under the projection $r_ p : \mathbf{P}(V) \to \mathbf{P}(W_ p)$. Since $\dim (W_ p) = \dim (V) - 1$, we see that $r_ p(B)$ has codimension $\geq 1$ in $\mathbf{P}(W_ p)$. For any $q \in \mathbf{P}(V)$ with $r_ p(q) \not\in r_ p(B)$ we see that the line $\ell = \overline{pq}$ connecting $p$ and $q$ works. $\square$
Lemma 43.23.5. Let $V$ be a vector space. Let $G = \text{PGL}(V)$. Then $G \times \mathbf{P}(V) \to \mathbf{P}(V)$ is doubly transitive.
Proof. Omitted. Hint: This follows from the fact that $\text{GL}(V)$ acts doubly transitive on pairs of linearly independent vectors. $\square$
Lemma 43.23.6. Let $k$ be a field. Let $n \geq 1$ be an integer and let $x_{ij}, 1 \leq i, j \leq n$ be variables. Then is an irreducible element of the polynomial ring $k[x_{ij}]$.
\[ \det \left( \begin{matrix} x_{11}
& x_{12}
& \ldots
& x_{1n}
\\ x_{21}
& \ldots
& \ldots
& \ldots
\\ \ldots
& \ldots
& \ldots
& \ldots
\\ x_{n1}
& \ldots
& \ldots
& x_{nn}
\end{matrix} \right) \]
Proof. Let $V$ be an $n$ dimensional vector space. Translating into geometry the lemma signifies that the variety $C$ of non-invertible linear maps $V \to V$ is irreducible. Let $W$ be a vector space of dimension $n - 1$. By elementary linear algebra, the morphism
\[ \mathop{\mathrm{Hom}}\nolimits (W, V) \times \mathop{\mathrm{Hom}}\nolimits (V, W) \longrightarrow \mathop{\mathrm{Hom}}\nolimits (V, V),\quad (\psi , \varphi ) \longmapsto \psi \circ \varphi \]
has image $C$. Since the source is irreducible, so is the image. $\square$
Let $V$ be a vector space of dimension $n + 1$. Set $E = \text{End}(V)$. Let $E^\vee = \mathop{\mathrm{Hom}}\nolimits (E, \mathbf{C})$ be the dual vector space. Write $\mathbf{P} = \mathbf{P}(E^\vee )$. There is a canonical linear map
\[ V \longrightarrow V \otimes _\mathbf {C} E^\vee = \mathop{\mathrm{Hom}}\nolimits (E, V) \]
sending $v \in V$ to the map $g \mapsto g(v)$ in $\mathop{\mathrm{Hom}}\nolimits (E, V)$. Recall that we have a canonical map $E^\vee \to \Gamma (\mathbf{P}, \mathcal{O}_\mathbf {P}(1))$ which is an isomorphism. Hence we obtain a canonical map
\[ \psi : V \otimes \mathcal{O}_\mathbf {P} \to V \otimes \mathcal{O}_\mathbf {P}(1) \]
of sheaves of modules on $\mathbf{P}$ which on global sections recovers the given map. Recall that a projective bundle $\mathbf{P}(\mathcal{E})$ is defined as the relative Proj of the symmetric algebra on $\mathcal{E}$, see Constructions, Definition 27.21.1. We are going to study the rational map between $\mathbf{P}(V \otimes \mathcal{O}_\mathbf {P}(1))$ and $\mathbf{P}(V \otimes \mathcal{O}_\mathbf {P})$ associated to $\psi $. By Constructions, Lemma 27.16.10 we have a canonical isomorphism
\[ \mathbf{P}(V \otimes \mathcal{O}_\mathbf {P}) = \mathbf{P} \times \mathbf{P}(V) \]
By Constructions, Lemma 27.20.1 we see that
\[ \mathbf{P}(V \otimes \mathcal{O}_\mathbf {P}(1)) = \mathbf{P}(V \otimes \mathcal{O}_\mathbf {P}) = \mathbf{P} \times \mathbf{P}(V) \]
Combining this with Constructions, Lemma 27.18.1 we obtain
43.23.6.1
\begin{equation} \label{intersection-equation-r-psi} \mathbf{P} \times \mathbf{P}(V) \supset U(\psi ) \xrightarrow {r_\psi } \mathbf{P} \times \mathbf{P}(V) \end{equation}
To understand this better we work out what happens on fibres over $\mathbf{P}$. Let $g \in E$ be nonzero. This defines a nonzero map $E^\vee \to \mathbf{C}$, hence a point $[g] \in \mathbf{P}$. On the other hand, $g$ defines a $\mathbf{C}$-linear map $g : V \to V$. Hence we obtain, by Constructions, Lemma 27.11.1 a map
\[ \mathbf{P}(V) \supset U(g) \xrightarrow {r_ g} \mathbf{P}(V) \]
What we will use below is that $U(g)$ is the fibre $U(\psi )_{[g]}$ and that $r_ g$ is the fibre of $r_\psi $ over the point $[g]$. Another observation we will use is that the complement of $U(g)$ in $\mathbf{P}(V)$ is the image of the closed immersion
\[ \mathbf{P}(\mathop{\mathrm{Coker}}(g)) \longrightarrow \mathbf{P}(V) \]
and the image of $r_ g$ is the image of the closed immersion
\[ \mathbf{P}(\mathop{\mathrm{Im}}(g)) \longrightarrow \mathbf{P}(V) \]
Lemma 43.23.7. With notation as above. Let $X, Y$ be closed subvarieties of $\mathbf{P}(V)$ which intersect properly such that $X \not= \mathbf{P}(V)$ and $X \cap Y \not= \emptyset $. For a general line $\ell \subset \mathbf{P}$ with $[\text{id}_ V] \in \ell $ we have
$X \subset U_ g$ for all $[g] \in \ell $,
$g(X)$ intersects $Y$ properly for all $[g] \in \ell $.
Proof. Let $B \subset \mathbf{P}$ be the set of “bad” points, i.e., those points $[g]$ that violate either (1) or (2). Note that $[\text{id}_ V] \not\in B$ by assumption. Moreover, $B$ is closed. Hence it suffices to prove that $\dim (B) \leq \dim (\mathbf{P}) - 2$ (Lemma 43.23.4).
First, consider the open $G = \text{PGL}(V) \subset \mathbf{P}$ consisting of points $[g]$ such that $g : V \to V$ is invertible. Since $G$ acts doubly transitively on $\mathbf{P}(V)$ (Lemma 43.23.5) we see that
\[ T = \{ (x, y, [g]) \mid x \in X, y \in Y, [g] \in G, r_ g(x) = y\} \]
is a locally trivial fibration over $X \times Y$ with fibre equal to the stabilizer of a point in $G$. Hence $T$ is a variety. Observe that the fibre of $T \to G$ over $[g]$ is $r_ g(X) \cap Y$. The morphism $T \to G$ is surjective, because any translate of $X$ intersects $Y$ (note that by the assumption that $X$ and $Y$ intersect properly and that $X \cap Y \not= \emptyset $ we see that $\dim (X) + \dim (Y) \geq \dim (\mathbf{P}(V))$ and then Varieties, Lemma 33.34.3 implies all translates of $X$ intersect $Y$). Since the dimensions of fibres of a dominant morphism of varieties do not jump in codimension $1$ (Varieties, Lemma 33.20.4) we conclude that $B \cap G$ has codimension $\geq 2$.
Next we look at the complement $Z = \mathbf{P} \setminus G$. This is an irreducible variety because the determinant is an irreducible polynomial (Lemma 43.23.6). Thus it suffices to prove that $B$ does not contain the generic point of $Z$. For a general point $[g] \in Z$ the cokernel $V \to \mathop{\mathrm{Coker}}(g)$ has dimension $1$, hence $U(g)$ is the complement of a point. Since $X \not= \mathbf{P}(V)$ we see that for a general $[g] \in Z$ we have $X \subset U(g)$. Moreover, the morphism $r_ g|_ X : X \to r_ g(X)$ is finite, hence $\dim (r_ g(X)) = \dim (X)$. On the other hand, for such a $g$ the image of $r_ g$ is the closed subspace $H = \mathbf{P}(\mathop{\mathrm{Im}}(g)) \subset \mathbf{P}(V)$ which has codimension $1$. For general point of $Z$ we see that $H \cap Y$ has dimension $1$ less than $Y$ (compare with Varieties, Lemma 33.35.3). Thus we see that we have to show that $r_ g(X)$ and $H \cap Y$ intersect properly in $H$. For a fixed choice of $H$, we can by postcomposing $g$ by an automorphism, move $r_ g(X)$ by an arbitrary automorphism of $H = \mathbf{P}(\mathop{\mathrm{Im}}(g))$. Thus we can argue as above to conclude that the intersection of $H \cap Y$ with $r_ g(X)$ is proper for general $g$ with given $H = \mathbf{P}(\mathop{\mathrm{Im}}(g))$. Some details omitted. $\square$
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