Finding nonzero global sections of a line bundle

Question

Let $V_0 = \{s \neq t\}$ and $V_1 = \{s \neq -t\},$ which together give an open cover of $\mathbb{P}^1.$ Let $L'$ be the line bundle defined by the transition functions $h_{01}: V_0 \cap V_1 \rightarrow GL_1(\mathbb{C})$ is $h_{01}([s:t]) = ((s+t)/(s-t))^2$ and $h_{10} : V_0 \cap V_1 \rightarrow GL_1(\mathbb{C})$ is $h_{10}([s:t]) = ((s-t)/(s+t))^2.$

What would global sections on $L'$ look like? The hint is to find functions $s_0: V_0 \rightarrow \mathbb{C}$ and $s_1:V_1 \rightarrow \mathbb{C}$ which patch together, but I don't understand what it means for functions to patch together.

What would the basis of $\Gamma(\mathbb{P}^1, L')$ look like?

Answer 1

The other answer has the general theory of functions patching together covered pretty well, but there's an important difference for line bundles specified in terms of transition functions. For a line bundle $L$ on a space $X$ given in terms of a collection of transition functions $h_{ij}$ for an open cover $U_i$ of $X$, a global section is equivalent to a family of sections $s_i\in L(U_i)$ for all $i$ so that $s_i = h_{ij} s_j$ on $U_i\cap U_j$.

In our case, there's only two open sets in our cover and $h_{01}=h_{10}^{-1}$ where both are defined, so we only have one equation to worry about: $s_0 = \frac{(s+t)^2}{(s-t)^2} s_1$ for $s_i\in L(U_i)$. So in order to find global sections we'll need to find sections $s_0,s_1$ satisfying this equation.

First we need to identify $L(U_i)$. As both $U_0$ and $U_1$ are isomorphic to copies of affine space and $L$ is trivial on each of them, we know that we'll get a polynomial algebra in one variable over $\Bbb C$ as $L(U_i)$. For $U_0$, we can take our coordinate for affine space to be $\frac{s+t}{s-t}$, and for $U_1$ we can take our coordinate to be $\frac{s-t}{s+t}$, so we have that $L(U_0)=\Bbb C[\frac{s+t}{s-t}]$ and $L_1=\Bbb C[\frac{s-t}{s+t}]$.

From looking at our equation, we see that $s_0$ should be a polynomial in $\frac{s+t}{s-t}$ of degree at most two and similarly $s_1$ should be a polynomial in $\frac{s-t}{s+t}$ of degree at most two. By writing $s_0=\frac{p(s-t,s+t)}{(s-t)^2}$ and $s_1=\frac{q(s-t,s+t)}{(s+t)^2}$ for homogeneous degree two polynomials $p,q$ , we see that our equation forces $p=q$ and so by specifying $p$ we can think of a global section as a homogeneous degree two polynomial in $s-t$ and $s+t$. So we can think of a basis for the homogeneous polynomials of degree two as a basis for the space of global sections of this line bundle.

Answer 2

Patch together means that the functions $s_0$ and $s_1$ agree on the intersection $V_0 \cap V_1$, so that they can be "glued" to give a section on all of $\mathbb{P}^1$. Let's compare to a standard topological result:

Fact 1 Let $X$ be a topological space with open sets $U_1, U_2 \subseteq X$ such that $U_1 \cup U_2 = X$. Let $Y$ be any topological space. If $f_1 : U_1 \to Y$ and $f_2 : U_2 \to Y$ are continuous functions such that $f_1|_{U_1 \cap U_2} = f_2|_{U_1 \cap U_2}$, then there is a unique continuous function $f : X \to Y$ such that $f|_{U_1} = f_1$ and $f|_{U_2} = f_2$.

Proof Sketch. Let $$f(x) = \begin{cases} f_1(x) : x \in U_1 \\ f_2(x) : x \in U_2 \end{cases}$$ Show that $f$ is well-defined and continuous, then that is the unique continuous function satisfying the desired property.

More generally, we have the following:

Fact 2 Let $X$ and $Y$ be topological spaces. Let $C$ be the presheaf of sets on $X$ defined by letting $C(U)$ be the set of continuous functions $U \to Y$ (with restriction maps $f \mapsto f|_V : C(U) \to C(V)$ whenever $V \subseteq U$). $C$ is a sheaf of sets on $X$.

The proof of Fact 2 is very similar to the proof of Fact 1 (which just verifies a special case of checking the equalizer condition for this presheaf). And there is another related fact:

Fact 3 Let $f : X \to Y$ be a continuous function between topological spaces. Let $D$ be the presheaf of sets on $Y$ defined by $D(U) = \{g : U \to X \mid g~\text{is continuous and } f \circ g = \operatorname{id}_U\}$. $D$ is a sheaf of sets on $Y$.

(Indeed, Fact 3 implies Fact 2 by the étale space construction for presheaves) In algebraic geometry, we have a very similar situation:

Fact 3' Let $f : X \to Y$ be a morphism of schemes. Let $C$ be the presheaf of sets on $Y$ defined by letting $C(U)$ be the set of morphisms $g : U \to X$ such that $f \circ g = \operatorname{id}_U$. $C$ is a sheaf of sets on $Y$.

The line bundle $L$ can be thought of (or defined as!) as a scheme morphism $f : L \to \mathbb{P}^1$ such that $\Gamma(U,L) = C(U)$ in the notation of Fact 3', which tells us that we're allowed to "glue" local sections of a line bundle together as long as they agree on the intersections of their domains of definition. In other words, $\Gamma(\mathbb{P}^1,L)$ is the equalizer of the maps $\Gamma(V_0,L) \times \Gamma(V_1,L) \rightrightarrows \Gamma(V_0 \cap V_1,L)$, with the first map being given by $(s_0, s_1) \mapsto s_0|_{V_0 \cap V_1}$ and the second map being given by $(s_0, s_1) \mapsto s_1|_{V_0 \cap V_1}$.