The Stacks project

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.

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 $r_ p|_ X : X \to \mathbf{P}(W_ p)$.

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.

  1. 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 $r_ p(x)$.

  2. 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

\[ Y \cap r_ p^{-1}(r_ p(Z)) = (Y \cap Z) \cup E \]

with $E \subset Y$ closed and $\dim (E) \leq \dim (Y) + \dim (Z) + 1 - n$.

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

\[ \det \left( \begin{matrix} x_{11} & x_{12} & \ldots & x_{1n} \\ x_{21} & \ldots & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots \\ x_{n1} & \ldots & \ldots & x_{nn} \end{matrix} \right) \]

is an irreducible element of the polynomial ring $k[x_{ij}]$.

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
\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

  1. $X \subset U_ g$ for all $[g] \in \ell $,

  2. $g(X)$ 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. $\square$

Comments (3)

Comment #9451 by AprilGrimoire on

Lemma 0B2T: did not exist in the conclusion of the statement.

Comment #9452 by AprilGrimoire on

Lemma 0B2V: I think here maybe its better to explain this element is geometrically irreducible, and reduce to an algebraically closed field so image could be considered at closed points, which are decent linear morphisms.

Comment #9457 by AprilGrimoire on

Lemma 0B2V: Irreducibility of polynomials corresponds to integralness of closed subschemes. Therefore, I think irreducibility in the proof concerning schemes should be replaced with integralness.

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