In this post I will contrast the definitions of line bundles in algebraic and differential geometry and give an overview of the most basic things you can do with line bundles. Line bundles in differential geometry are perhaps easier to understand and visualize, so it should provide some intuition of what goes on in the algebraic setting.
Line Bundles in Differential Geometry
First let’s rephrase the common definition of line bundle from differential geometry in the language of sheaves.
Let \(X\) be a (topological, smooth, analytic, etc.) manifold. Roughly speaking, a real line bundle over \(X\) is a manifold \(E\) together with a map \(\pi : E \to X\) such that each fiber \(\pi^{-1}(x)\) is a \(1\)-dimensional vector space. Furthermore, \(\pi\) is locally trivial, meaning each point \(x \in X\) has an open neighbourhood \(U\) such that \(\DeclareMathOperator{\RR}{\mathbb{R}}\pi^{-1}(U) \cong U \times \RR\) with appropriate compatibility conditions. Giving such a local trivialization is equivalent to saying that \(X\) is locally isomorphic to the trivial line bundle \(U \times \RR \to U\).
Let \(\DeclareMathOperator{\cO}{\mathcal{O}}\cO_X\) denote the sheaf of (an appropriate class of) real-valued functions on \(X\). Let \(\DeclareMathOperator{\cE}{\mathcal{E}}\cE\) denote the sheaf of sections of a line bundle \(E\). Note that \(\cE\) is an \(\cO_X\)-module, under pointwise multiplication. The global sections of the trivial line bundle \(\pi : X \times \RR \to X\) correspond to those (continuous, smooth, analytic etc.) maps \(X \to \RR\). In this way, a local trivialization as above gives an isomorphism of \(\cO_X \vert_U\)-modules \(\cE \vert_U \cong \cO_X \vert_U\) and conversely.
If \(s : X \to E\) is a global section of a line bundle \(E\), it makes sense to ask if \(s\) vanishes at some point \(x \in X\), and we can check this on any local trivialization of \(E\) near \(x\). We call the subset of points of \(X\) where \(s\) vanishes the vanishing locus of \(s\). It is clearly a closed subspace. We denote its complement by \(X_s\) and call it the non-vanishing locus. \(s\) gives a trivialization of \(X_s\): we have a map \(X_s \times \RR \to \pi^{-1}(X_s)\) sending \((x, \lambda) \mapsto \lambda \cdot s(x)\). This is an isomorphism as we can ‘divide’ by a non-zero vector \(s(x)\) in a 1-dimensional vector space, in other words, ask what scalar multiple of \(s(x)\) any other vector is.
Given two local trivailizations of \(E\) say \(\phi_U : \pi^{-1}(U) \to U \times \RR\) and \(\phi_V : \pi^{-1}(V) \to V \times \RR\), we can ask for the transition functions between these: \(\phi_V \circ \phi_U^{-1} : (U \cap V) \times \RR \to (U \cap V) \times \RR\) is of the form \((x, v) \mapsto (x, T(x) v)\). We call \(T\) the transition function from \(U\) to \(V\). For each \(x \in U \cap V\), \(T(x)\) is an \(\RR\)-linear isomorphism, thus \(T\) is an element of \(\mathrm{GL}_1(\cO_X(U \cap V))\).
Line Bundles in Algebraic Geometry
We will now explain the analogues of things mentioned in algebraic geometry. Here we will work with sheaves of modules. A line bundle on a scheme \(X\) is a locally free sheaf \(\DeclareMathOperator{\cL}{\mathcal{L}}\cL\) of rank 1, i.e. there is an open cover of \(X\) such that for each \(U\) in the cover, \(\cL \vert_U \cong \cO_X \vert_U\) as \(\cO_X \vert_U\)-modules. In particular, \(\cL\) is quasicoherent. As apposed to the above, we forgo defining the space associated to this sheaf and simply work with the sheaf itself.
Given a global section \(s \in \cL(X)\), we can again ask if \(s\) vanishes at some point \(x \in X\). \(s\) vanishing at \(x\) simply means that if \(\cL \vert_U \cong \cO_X \vert_U\) is a local trivialization, then \(s_p\) maps into the maximal ideal of \(\cO_{X,p}\) under \(\cL_p \to \cO_{X, p}\). It is easy to see that this is a well-defined notion. We define the non-vanishing locus \(X_s\) of \(s\) exactly as before, which is again an open subset. Again, \(s\) gives a trivialization of \(X_s\). It is perhaps a good exercise to spell out the details of this: Suppose \(\DeclareMathOperator{\Spec}{Spec}U = \Spec A \subseteq X_s\) is an affine open subset which trivializes \(\cL\), say via \(\phi_U : \cL \vert_U \xrightarrow{\sim} \cO_X \vert_U\). The restriction of \(s\) to \(U\) gives an element \(\phi_U(s \vert_U) \in A\), which does not vanish on \(\Spec A\), hence is a unit. Define \(\tilde{\phi}_U : \cL \vert_U \to \cO_X \vert_U\) by \(t \mapsto \phi_U(t) / \phi_U(s \vert_U)\). These glue to give an isomorphism \(\cL \vert_{X_s} \to \cO_X \vert_{X_s}\), which we will suggestively call \(\cdot / s\).
Given two global sections \(s, s'\) of \(\cL\), what we have done above tells us that \(s' / s\) makes sense as a section of \(\cO_X \vert_{X_s}\). In fact, multiplying by \(s' / s\) is precisely the transition function from \(X_{s'}\) to \(X_s\) with our trivializations as constructed above.
Line Bundles on \(\DeclareMathOperator{\PP}{\mathbb{P}}\PP^n\)
The first non-trivial (and important) example one might see is the line bundle \(\cO(1)\) on \(\PP^n\). I will now try to motivate its construction.
Classically, \(\PP^n\) say over a field \(k\), is defined as the quotient of \(k^{n+1} \setminus \{0\}\) modulo the equivalence relation \(v \sim \lambda v\) for \(\lambda \in k^\times\). Thus each point in \(\PP^n\) has homogeneous coordinates \([x_0, \ldots, x_n]\) that are defined up to scaling. While \(x_0\) on its own does not give a well-defined function, it does make sense to ask for the set of points where \(x_0\) vanishes, or more generally where homogeneous polynomials in the \(x_i\) vanish. The \(x_i\) are not global sections of \(\cO_{\PP^n}\) itself, however they do form a line bundle, called \(\cO(1)\). Let’s explain how:
We want to have all the \(x_i\) as global sections of \(\cO(1)\). What should the sections on \(D_+(x_i)\) look like? \(\cO(1)(D_+(x_i))\) should be a \(k[x_0/x_i, \ldots, x_n/x_i]\)-module that contains \(x_0, \ldots, x_n\). We see that it is enough to contain \(x_i\), and thus we should define it to be \(\cO(1)(D_+(x_i)) = k[x_0/x_i, \ldots, x_n/x_i] \cdot x_i\), where we can think of the rightmost \(x_i\) as a formal symbol for a basis element. What is the transition function of \(\cO(1)\) from \(D_+(x_i)\) to \(D_+(x_j)\)? We have \(1 \cdot x_i = (x_i / x_j) \cdot x_j\). Thus the transition function with respect to our chosen basis (\(x_i\) and \(x_j\) respectively) is multiplication by \(x_i / x_j\). Compare this with the transition function for \(X_{s'}\) to \(X_s\) above!
Suppose now we are given sections \(s_0, \ldots, s_n\) of a line bundle \(\cL\) on a scheme \(X\). Suppose further that the \(s_i\) do not simultaneously vanish on \(X\). Our intuition for projective space tells us that we should be able to define a morphism \(f = [s_0, \ldots, s_n] : X \to \PP^n\) using these sections – what the \(s_i\)’s are don’t matter, what matters is that their ratios make sense. We do this by gluing together morphisms \(f_i : X_{s_i} \to D_+(x_i)\). To define \(f_i\), it suffices to give a homomorphism \(k[x_0/x_i, \ldots, x_n/x_i] = \Gamma(D_+(x_i), \cO_{\PP^n}) \to \Gamma(X_{s_i}, \cO_X)\). Of course, we send \(x_j / x_i\) to \(s_j / s_i\), and check that these do indeed glue to give a morphism \(f\). As remarked in the previous paragraph, the transition functions of \(\cL\) and the pullback \(f^*(\cO(1))\) from \(X_{s_i}\) to \(X_{s_j}\) are the same. Thus \(\cL \cong f^*(\cO(1))\) as line bundles.
Conversely, if we have a morphism \(f : X \to \PP^n\). We get a line bundle \(\cL = f^*(\cO(1))\) and sections \(s_i = f^*(x_i)\) of \(\cL\) which do not simultaneously vanish on \(X\). This gives a bijection between morphisms \(f : X \to \PP^n\) and line bundles \(\cL\) on \(X\) with global sections \(s_0, \ldots, s_n\) not simultaneously vanishing on \(X\), up to isomorphism of this data. From this we can study morphisms to projective space via studying line bundles.
References
[1] Ravi Vakil, Foundations of Algebraic Geometry. Available here.