Lecture notes from Karl Christ’s Algebraic Curves course, UT Austin, Spring 2024. Live-TeXed during lecture; any errors are mine.
\(X\) is still a smooth curve over an algebraically closed field \(K\).
Brief Follow-up from Previous Class
We discussed “finite type” vs “finite” last class. A map \(f:x\to Y\) is
quasi finite if it has finitely many points in each fiber
finite if there exists an affine cover \(U_\alpha\) of \(Y\) such that for
\begin{align*} f|_{f^{-1}(U_\alpha)}: f^{-1}(U_\alpha) \to U_\alpha \end{align*}the corresponding inclusion \(A\rightarrow B\) realizes \(B\) as a finitely generated \(A\)-module; where \(U_\alpha = \Spec A\) and \(f^{-1}(U_\alpha) = \Spec B\).
Theorem
A map as above is finite if and only if it is quasi-finite and proper.
Example
Take the inclusion \(\mathbb A^1 \setminus \{0\} \to \mathbb A^1\). It is not finite type because \(K[x,x^{-1}]\) is not finitely generated as a module over \(K[x]\).
Line bundles and linear series
We ended last class by defining the sheaf \(\mathcal O_X(D)\) to be the sheaf of rational functions \(f\) of \(X\) such that \(\div (f) + D\) is effective.
Example
\(\mathcal O_{\mathbb P^1}(2P - 3Q)\) is the sheaf of regular functions which have a zero of order 3 or greater at \(Q\), a pole of order at most \(2\) at \(P\) and no poles anywhere else on \(\mathbb P^1\).
We have the following facts immediately.
- \(\mathcal O_X(D) \cong \mathcal O_X(D') \iff D\sim D'\)
- Conversely, any invertible sheaf \(L\) is of the form \(\mathcal O_X(D)\) for some divisor \(D\)
We call \(L\) effective if there exists \(D\) effective such that \(L\cong \mathcal O_X(D)\) which happens if and only if \(h^0(X,L) > 0\). Note that there are effective sheaves which are not invertible in general.
Linear series
The degree of a line bundle \(L\cong \mathcal O_X(D)\) is the degree of \(D\). Furthermore, \(\mathcal O_X(D + D') = \mathcal O_X(D)\otimes \mathcal O_X(D')\). We can now define linear series, objects which classify maps \(X\to \mathbb P^r\).
Definition
A linear series is a subspace \(V\) of \(H^0(X,L)\).
- It has degree equal to the degree of \(L\).
- It has rank equal to \(\dim(V) - 1\).
- The linear series is called complete if \(V = H^0(X,L).\)
- A base point of a linear series \((L,V)\) is a point \(P\) such that \(\dim(V) = \dim(V(-P))\).
A linear series of degree \(d\) and rank \(r\) is called a \(g^r_d\).
Why do we care about linear series? It turns out they classify maps to \(\mathbb P^r\).
Given a \(g^r_d\) \((L,V)\) that is base point free, we obtain a map \(\varphi_V:X\to \mathbb P^r\) given by
\begin{align*} P\mapsto \{s ~\mid~ s(P) = 0\} \in \mathbb PV^* \cong \mathbb P^r. \end{align*}Note that \(V^*\) is the dual space here. This is the coordinate free version, but if we choose a basis \(s_0,...,s_r\) of \(V\) (remember that \(V\) is \(r+1\) dimensional) then we can write this map more concretely as
\begin{align*} P\mapsto [s_0(P): ... : s_r(P)]. \end{align*}A question was asked: why does \(\{s\in V ~\mid~ s(P) = 0\}\) specify an element in the dual space of \(V\)? It’s because the condition \(s(P) = 0\) specifies a hyperplane in \(V\), a subspace of codimension 1, and hence is given by an element of the dual space.
In the situtation \(\varphi_V:V\to \mathbb P^r\) above, we have \(L\cong \varphi_V^*\mathcal O_{\mathcal P^r}(1)\); in other words, we can recover \(L\) from the map \(\varphi_V\). To summarize:
Proposition
There is a bijective correspondence between maps \(f:X\to \mathbb P^r\) up to the action of \(\operatorname{PGL}_{r + 1}\) with non-degenerate image and a base-point free linear series.
Notes: By non-degenerate we just mean that the image of \(f\) above isn’t contained in any linear subspace of \(\mathbb P^r\). This isn’t the case with the map \(\varphi_V\) we defined above – if it were, then \(s_0(P), ...,s_r(P)\) would have a nontrivial linear relation, which isn’t the case since they were chosen to be a basis. If \(f:X\to \mathbb P^r\) did have image lying in some linear subspace, then we’d simply decrease the value of \(r\).
Remark
If \(D\in \operatorname{Div}(X)\) is the base locus of \(V\), then the map \(\varphi_V\) is defined to be the map induced by \(V(-D)\).
Remark
If the \(g^r_d\) \((V,L)\) has degree \(d\) then
\begin{align*} d = \deg(\varphi_V) \cdot \deg(\varphi_V(X)) + \deg(D). \end{align*}Ampleness
\(L\) is called very ample if \(\varphi_L\) is an embedding. \(L\) is called ample if \(L^{\otimes n}\) is very ample for some \(n > 0\).
Proposition
\(L\) is very ample if and only if it is effective and
- \(h^0(X,L(-P)) = h^0(X,L) - 1\) for all \(P\in X\)
- \(h^0(X,L(-P - Q)) = h^0(X.L) - 2\) for all \(P,Q\in X\).
The example we’ll discuss next time to have in mind is the following:
Example
\(\mathbb P^1 \to \mathbb P^3\), \((s,t) \mapsto (s^3 : st^2: t^3)\).