In algebraic geometry, the first step-the construction step is often the most difficult step. It is often painful for me to type such things, after writing on paper twice. They are difficult and very easy to forget. But with one step more, some fruitful results will arise, and they make me very surprised. I have to admit that I understand nothing without doing examples and see how to apply such general constructions in specific contexts. On these parts, we will construct line bundles, invertible sheaves on smooth projective curves, and their relations to the Picard group, and sheaf cohomology, which is a powerful tool to compute things explicitly.
1. Line bundles. For simplicity, we always denote $X$ a smooth projective curve on the ground field $\mathbb{C}$, but we can easily generalize these definitions to any scheme over any field.
Definition 1.1. Let $L$ be a set and $\pi: X\to L$ a map. A line bundle chart for $X$ over $L$ is a pair $(U,\phi)$ where $U$ is an open subset of $X$, and $\phi: \pi^{-1}(U)\to \mathbb{C}\times U$ is a bijection map such that $pr_2\circ\phi=\pi$ on $U$, where $pr_2$ is a projection map from $\mathbb{C}\times U\to U$.
Let us give an example. If $L:=\mathbb{C}\times X$ and $\pi: \mathbb{C}\times X\to X$ is a projection map. We then take any open subset $U$ of $X$, then $\pi^{-1}(U)=\mathbb{C}\times U$, and $\phi$ the identity map. Then it can be seen that $(U,\phi)$ is a line bundle chart for $X$ over $L$.
One can se directly from the definition that for all $p\in U$, the fiber of $p$ is of the form $\mathbb{C}\times \{p\}$, which is a one-dimensional vector space. And we can define the $\mathbb{C}$-vector space structure on the fiber. Once we have the chart at hand, we want our charts covers $X$, and they are compatible in some ways, so that we can glue them and obtain a line bundle structure on $X$.
Definition 1.2. Let $(L,\pi)$ be defined as above, and $(U_1,\pi_1), (U_2,\pi_2)$ are two line bundle charts on $X$ over $L$, then $(U_1,\pi_1), (U_2,\pi_2)$ is called compatible if for all $p\in U_1\cap U_2$, and $v\in \mathbb{C}$, $\pi_2\circ\pi_1^{-1}(v,p)=(f(p)v,p)$, where $f\in \mathscr{O}(U_1\cap U_2)$ and $f$ is nowhere zero in $U_1\cap U_2$.
If we fix $p\in U_1\cap U_2$, and let $v$ varies on $\mathbb{C}$, we then actually obtain the linear isomorphism between fibers of $p$ via two charts. Let us come back with an example above, it can be seen that for any $(U_1,id_1), (U_2,id_2)$ are two line bundle charts for $X$ over $L$, then on $U_1\cap U_2$, the map $id_2\circ id_1^{-1}$ is just the identity map, which is for sure satisfied the Definition 1.2. This yields they are compatible with each other.
Definition 1.3. Let $(L,\pi)$ be defined as above, then $(L,\pi)$ is called line bundle for $X$ over $L$ if there exists an open cover $\{U_i\}_i$ of $X$, and each $(U_i,\pi_i)$ is line bundle charts, which are compatible with each other.
One can see from this that if $L:=\mathbb{C}\times X$ and $\pi: L\to X$ is the projection map, then $(L,\phi)$ is a line bundle on $X$, which is called trivial line bundle on $X$.
Important remark. As we have seen, if $(L,\pi)$ is a line bundle on $X$, and $(U_i,\pi_i)$ are line bundle charts on $X$, with $\{U_i\}$ is an open cover of $X$, then if we let $t_{ij}:=\pi_j\circ\pi_i^{-1}$, then $t_{ij}$ satisfies the cocycle condition:
1. $t_{ii}=1$
2.$ t_{ij}=t_{ji}^{-1}$ on $U_i\cap U_j$.
3. $t_{ik}=t_{ij}\circ t_{jk}$ on $U_i\cap U_j\cap U_k$.
We call $t_{ij}$ the transition functions. Transition functions of line bundles play an important role in the similarity between line bundles and the first cohomology group, as we will see later.
2. Tautological line bundle. One of the most important construction of line bundle is the tautological line bundle. Let $X$ be a smooth projective curve (again, one can do this without the assumption "curve"), and $\Phi: X\to \mathbb{P}^n$ a morphism, then for all $p\in X$, we have $\Phi(p)=(f_0(p):...:f_n(p))$, where $f_0,...,f_n$ do not vanish vanish identically at any point $p\in X$.
The projective space $\mathbb{P}^n$ can be identical with a set of one-dimensional subspaces of $\mathbb{C}^{n+1}$, i.e. if $(x_0:...:x_n)\in\mathbb{P}^n$, then we can identical this point as a subspace of $\mathbb{C}^{n+1}$ generated by $(x_0,...,x_n)$. Then for any $p\in X$, $\Phi(p)$ is a one-dimensional subspace of $\mathbb{C}^{n+1}$.
We then denote $L:=\{(v,p)|v\in \Phi(p), p\in X\}$, and $\pi: L\to X$ is a projection map $(v,p)\mapsto p$. Let $U_i\subset \mathbb{P}^{n}$ be an open subset of $\mathbb{P}^n$ that consist all points of $\mathbb{P}^n$ with the $i$-th coordinate is non-zero. Then $V_i:=\Phi^{-1}(U_i)$ is open cover of $X$, which consists of all point $p\in X$ such that $f_i(p)\ne 0$.
Line bundle charts $(V_i,\pi_i)$ can be constructed as follows. Let $\pi_i: \pi^{-1}(V_i)\to \mathbb{C}\times V_i$ sending $(v,p)$ to $(v_i,p)$ where $v_i$ is the $i$-th coordinate of $V$. We will prove that $\pi_i$ is a bijection map. For $p\in U_i$, it can be seen $v\in \Phi(p)$, and $v_j=\lambda f_j(p)$, where $\lambda\in \mathbb{C}$, for all $j$. In particular, $v_i=\lambda f_i(p)$, hence $v_j=v_i (f_i(p)/f_i(p))$ for all $j$. Hence $\pi_i^{-1}$ is given by $(v_i,p)\mapsto (v,p)$, where $v=(v_if_0(p)/f_i(p),...,v_if_n(p)/f_i(p))$. And $\pi_i$ is a bijection. Also, one can see that $pr_2(\pi_i(v,p))=p=\pi(v,p)$, and $pr_2\circ\pi_i=\pi$. Hence, $(V_i,\pi_i)$ is line bundle charts on $X$.
We need to check the compatible from this construction, so that $(L,\pi)$ is a line bundle over $X$. But then, it can be seen that
$$\pi_j\circ\pi_i^{-1}(v_i,p)=\pi_j((v_if_0(p)/f_i(p),...,v_if_n(p)/f_i(p)),p)=(\frac{f_j}{f_i}(p)v_i,p)$$
$$\pi_j\circ\pi_i^{-1}(v_i,p)=\pi_j((v_if_0(p)/f_i(p),...,v_if_n(p)/f_i(p)),p)=(\frac{f_j}{f_i}(p)v_i,p)$$
And the function $f_j/f_i$ is regular and nowhere zero on $U_i\cap U_j$. Hence, $(L,\pi)$ is a line bundle on $X$. This is called the tautological line bundle.
Remark. One may know that we care about the case of curve, because for curves, we can construct a morphism from curves to $\mathbb{P}^n$ conveniently via divisors. And the tautological line bundle constructed above can be associated with divisors on curves.
3. Rational sections of line bundles. We now connect the notions of line bundles and divisors on curves. This can be seen via the rational sections of line bundles.
Definition 3.1. Let $(L,\pi)$ be a line bundle on an irreducible smooth projective curve $X$. Let $U$ be any open subset of $X$, a rational section of $L$ is a map from $s: U\to L$, where $U$ is an open subset of $X$, such that
1. $\pi\circ s=id_U$
2. For any line bundle chart $(V,\phi)$ for $X$ over $L$, the map $pr_1\circ \phi\circ s: U\cap V\to\mathbb{C}$ is a rational function.
Note that for any irreducible smooth projective curve, any open subset is open. Hence, if $pr_1\circ \phi\circ s$ is a rational function on $U\cap V$, it is a rational function on $X$. But this function is not unique, since it depends on the choice of the line bundle chart. If we choose another line bundle chart $(V',\phi')$, then due to the compatible conditions, we have $\phi'\circ\phi^{-1}(v,p)=(f(p)v,p)$, where $f\in \mathscr{O}(V\cap V')$, and $f$ is nowhere zero in $V\cap V'$.
Let $p\in U\cap V\cap V'$, consider $l:=s(p)$, which is in $\pi^{-1}(p)$, i.e. $\pi(l)=p$. Due to the definition of a line bundle chart, we have $pr_2\circ\phi(l)=\pi(l) = p$, i.e. $\phi(l)$ is of the form $(v,p)$ for some $v\in\mathbb{C}$. And hence, $pr_1\circ\phi\circ s(p)=v$. Furthermore, due to the compatible condition, one has $\phi'\circ\phi^{-1}(v,p)= (f(p)v,p) = \phi'(l)$, i.e. $pr_1\circ\phi'\circ s(p) = f(p)v$, where $f$ is a regular nowhere zero function on $V\cap V'$.
This implies that if $s$ is a rational sections of $L$ on $U$, and $(U_i,\pi_i)$ are line bundle charts cover $X$. Then $pr_1\circ\phi_i\circ s=t_{ij}.pr_1\circ\phi_j\circ s$ on $U\cap U_i\cap U_j$. This gives rise to define the divisor of a rational section.
Definition 3.2. Let $s$ be a rational section of a line bundle $(L,\pi)$. For any line bundle chart $(V,\phi)$, we define $div(s):=div(pr_1\circ\phi\circ s)$.
From a nowhere zero condition of transition functions, we can see that this definition is well-defined. More important, it gives us a way to connect line bundles and divisors.
Proposition 3.3. Let $s_1,s_2$ be rational sections of a line bundle $(L,\pi)$, then $div(s_1)\sim div(s_2)$.
Proof. Assume that $s_1,s_2$ are maps from $U_1\to L$ and $U_2\to L$, respectively. If we fix any line bundle chart $(U,\phi)$, and denote $\alpha_i:=pr_1\circ \phi\circ s_i$, then $div(s_1)-div(s_2)=div(\alpha_1/\alpha_2)$. We will prove that the $\alpha_1/\alpha_2$ does not depend on the choice of line bundle charts.
On the other hand, if $(V,\phi')$ is another line bundle chart, we define $\beta_i:=pr_1\circ\phi'\circ s_i$, then on $U_1\cap U\cap V$, we have $\beta_i=t\alpha_i$, where $t$ is a transition function, i.e. regular nowhere zero function on $U\cap V$, and hence, $\beta_1/\beta_2=\alpha_1/\alpha_2=:g$.
This yields $div(s_1)-div(s_2)(p)=ord_p(g)$ for all $p\in U\cap V$. Because such $(U,\phi)$ cover $X$, this holds for all $p\in X$. Hence, $div(s_1)\sim div(s_2)$ (Q.E.D)
A line bundle is characterized by the divisor of its rational sections. If we denote $LB'(X)$ the sets of all rational sections on all line bundle over $X$, then the map $LB'(X)\to Pic(X)$ sending $s_L$ to $div(s_L)$, where $s_L$ is a rational section of a line bundle $L$ is well-defined. But this is not enough for our purpose, we want to define $LB(X)$ as $LB'(X)$ modulo isomorphism of line bundles, so that $LB(X)$ is in bijection with $Pic(X)$. The detail of the isomorphism between line bundles will be discussed later.
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