How do we describe maps of line bundles on $\mathbb{P}^1$?

Solution 1:

My question: how can I describe, fiberwise, this homomorphism $\mathcal{O}(n)\rightarrow \mathcal{O}(m)$ in terms of the degree-$(m-n)$ polynomial, when $n<0$?

Answer: let $C$ be the projective line. There is by Hartshorne, Ex II.5.1 isomorphisms

$$Hom_{\mathcal{O}_C}(\mathcal{O}(m), \mathcal{O}(n)) \cong Hom_{\mathcal{O}_C}(\mathcal{O}_C, Hom(\mathcal{O}(m), \mathcal{O}(n))) \cong $$

$$Hom_{\mathcal{O}_C}(\mathcal{O}_C, \mathcal{O}(m)^*\otimes \mathcal{O}(n))\cong Hom_{\mathcal{O}_C}(\mathcal{O}_C, \mathcal{O}(n-m)) \cong H^0(C, \mathcal{O}(n-m)).$$

Here you should check that to give a map of $\mathcal{O}_C$-modules

$$s: \mathcal{O}_C \rightarrow \mathcal{O}(n-m)$$

is equivalent to give a global section of $\mathcal{O}(n-m)$ which is equivalent to giving a homogeneous polynomial of degree $n-m$ in $x_0,x_1$.

If $n-m < 0$ there are no such maps. If $n-m \geq 0$ the vector space of such maps has dimension $n-m+1$, and given a map $s$ it corresponds 1-1 to a homogeneous polynomial

$$s(x_0,x_1) \in \mathbb{C}[x_0,x_1]^{n-m}.$$

Example: A map $s:\mathcal{O}(m) \rightarrow \mathcal{O}(n)$

corresponds 1-1 to a homogeneous polynomial $s(x_0,x_n)$ of degree $n-m$ from the following argument:

We get a map of graded modules

$$ \phi: \mathbb{C}[x_0,x_1](m)\rightarrow \mathbb{C}[x_0,x_1](n)$$

defined by

$$\phi(f(x_0,x_1)):=s(x_0,x_1)f(x_0,x_1).$$

Sheafifying this map we get an induced map

$$s:\mathcal{O}(m) \rightarrow \mathcal{O}(n)$$

of $\mathcal{O}_C$-modules. This construction gives all such maps by the above argument.

Example: For any $n,m$ with $n-m \geq 0$ you get a vector space of dimension $n-m+1$ of such maps.

The fiber: The induced map at the fiber at $x:=(t-\alpha) \in C(\mathbb{C})$: Let $t:=x_1/x_0$. We get the following map $\phi_x$ between fibers $\mathcal{O}(m)(x)\cong \mathbb{C}x_0^m:=\mathbb{C}[t]/(t-\alpha)x_0^m$ and $\mathcal{O}(n)(x)$:

$$\phi_x: \mathbb{C}x_0^m \rightarrow \mathbb{C}x_0^n$$

defined by

$$\phi(x)(ux_0^m):=g(\alpha)ux_0^n$$

where

$s(x_0,x_1)=g(t)x_0^{n-m}$.

Here the point $x:=(a_0:a_1)$ has $a_0 \neq 0$ and by definition $x=(1: a_1/a_0)=(1:\alpha)$. There is an isomorphism

$$\mathcal{O}(m)(D(x_0)) \cong \mathbb{C}[t]x_0^m$$

and the point $x$ corresponds 1-1 to the maximal ideal $(t-\alpha) \subseteq \mathbb{C}[t]$ with $t:=x_1/x_0$. The fiber at $x$ is by definition

$$\mathcal{O}(m)(x)\cong \mathbb{C}[t]/(t-\alpha) x_0^m \cong\mathbb{C}x_0^m.$$

A generalization: Similar results hold for any projective scheme $X \subseteq \mathbb{P}^n_{A}$: There are isomorphisms

$$Hom_{\mathcal{O}_X}(\mathcal{O}(m), \mathcal{O}(n)) \cong Hom_{\mathcal{O}_X}(\mathcal{O}_X, \mathcal{O}(n-m))$$

$$ \cong H^0(X, \mathcal{O}(n-m)).$$

Hence for projective space $\mathbb{P}^n_k$ you get an equality

$$Hom_{\mathcal{O}}(\mathcal{O}(m), \mathcal{O}(n)) \cong k[x_0,..,x_n]^{n-m}.$$

Hence a map of sheaves

$$s: \mathcal{O}(m) \rightarrow \mathcal{O}(n)$$

is in 1-1 correspondence with a homogeneous polynomial

$$s(x_0,..,x_n) \in \mathbb{C}[x_0,..,x_n]^{n-m}$$

of degree $n-m$.

Note: Usually we write $k[x_i]_d$ instead of $k[x_i]^d$. The notation

$$k[x_i]^d :=\oplus_{i=1}^d k[x_i]e_i$$

usually means "the direct sum of $d$ copies of $k[x_i]$".

Note moreover: There is an exercise (Ex.II.5.9 in Hartshorne) classifying coherent sheaves on a projective scheme $X:=Proj(S) \subseteq \mathbb{P}^n_A$ with $A$ a finitely generated algebra over a field $k$. Assume $S$ is generated by $S_1$ as $A$-algebra and for a given graded $S$-module $M$, let

$$M_{\geq d}:=\oplus_{n \geq d} M_n.$$

Two graded $S$-modules $M,N$ are equivalent iff there is an integer $d\geq 0$ and an isomorphism $M_{\geq d} \cong N_{\geq d}$. A module $M$ is "quasi finitely generated" iff it is equivalent to a finitely generated module. The functor $F(M):=\tilde{M}$ gives an equivalence between the category of coherent sheaves on $X$ and the category of quasi finitely generated $S$-modules modulo this equivalence.

Example: In particular if $A:=\mathbb{C}$ is the field of complex numbers and $E$ is a coherent analytic sheaf on $X$, there is a quasi finitely generated $S$-module $M$ and an "isomorphism" $E \cong \tilde{M}^s$ where $(-)^s$ is the analytification functor described in

Smooth algebraic varieties are complex manifolds

Hence if $L^s, E^s$ are coherent analytic sheaves on $X$ with $L^s$ a line bundle, there is an isomorphism

$$Hom_{\mathcal{O}_X}(L,E) \cong Hom_{\mathcal{O}_X}(\mathcal{O}_X, E\otimes L^*) \cong H^0(X, E\otimes L^*) \cong $$

$$Hom_{\mathcal{O}_X^s}(\mathcal{O}_X^s, E^s \otimes (L^*)^s) \cong H^0(X^s, E^s\otimes (L^*)^s).$$

Hence you can calculate the maps from $L^s$ to $E^s$ using $L$ and $E$. Hence if $L^s,E^s$ are coherent analytic sheaves on $X^s$ and if $L^s$ is invertible, the set of maps $L^s \rightarrow E^s$ is finite dimensional: Since $E^s\otimes (L^*)^s$ is coherent it follows from HH.II.5.19:

The vector space

$$\Gamma(X^s, E^s\otimes (L^*)^s):=H^0(X^s, E^s\otimes (L^*)^s)\cong H^0(X, E\otimes L^*)$$

is finite dimensional. This is Theorem AppB.2.1 in Hartshorne.