Lecture notes from Karl Christ’s Algebraic Curves course, UT Austin, Spring 2024. Live-TeXed during lecture; any errors are mine.
Organizational Matters
Instructor: Karl Christ
References:
- Harshorne Chapter IV
- Arbasello, Corualla, Griffiths, Harris: Geometry of Algebraic Curves, Vol I (maybe Vol II also)
- Eisenbud and Harris: The practice of algebraic curves (Draft)
- Haris: Basic algebraic geometry for constructions of things like \(\mathbb Gr(k,V)\) in terms of coordinates, where you can really get your hands dirty.
Grade: There is no regular homework. There is a final exam, which is an oral exam. You can get extra credit by handing in up to 2 papers about 5-10 pages each. Topic can be anything from the class.
Office Hours: 15:30 - 17:00 on Monday. Can also talk after class on Wednesday at department coffee.
Overview of Algebraic Curves
The most natural (first) way to study algebraic curves is by interpreting them as the zero sets of polynomials. These are embedded algebraic curves. These have been studied almost since the inception of mathematics, the Ancient Greeks studied them for instance. We will typically interpret embedded algebraic curves as subsets of \(\mathbb P^r\) or more concretely in \(\mathbb P^r_{\mathbb C}\).
We will also study abstract curves i.e. one-dimensional schemes i.e. compact 1-dimensional complex manifolds (Riemann surfaces).
The main theme of this course will be the study of an algebraic curve \(C\) together with a map \(C\to \mathbb P^r_{\mathbb C}\). This allows us to study both abstract and embedded curves simultaneously.
Perhaps the first thing to not is that this adds nothing new to the theory.
Remark
Every abstract smooth curve over \(\mathbb C\) can be embedded in \(\mathbb P^3\).
Note that this fails for singular curves; the dimension of the Zariski tangent space can be arbitrarily large at a singularity and any embedding of a curve must map to a space whose dimension is at least as large as this dimension. You can find singularities whose Zariski tangent space is of arbitrarily large dimension.
The map \(\varphi:C\to \mathbb P^r\) is given by a line bundle \(L\) on \(C\) and a vector subspace \(V\) of \(H^0(C,L)\) of dimension \(r+d\), where \(d\) is the degree of \(\varphi\).
Genus
We thus far have two invariants: \(r\) the dimension of the ambient projective space \(\mathbb P^r\) and \(d\) the degree of the embedding \(\varphi:C\to \mathbb P^r\). The third primary invariant we consider is the genus \(g\) of \(C\to \mathbb P^r\). A natural question is this: given \(d\) and \(r\), what are the possible values of \(g\)?
Definition
Let \(X\) be a curve.
- The arithmetic genus of \(X\) is \(p_a(X) = 1 - P_X(0)\) where \(P_X\) is the Hilbert polynomial of \(X\).
- The geometric genus of \(X\) is \(p_g(X) = \dim_k\Gamma(X,\omega_X)\) where \(\omega_X\) is the canonical sheaf \(\omega_X = \bigwedge_{i=1}^n \Omega_X\).
These two notions agree for curves and are equal to \(H^1(X,\mathcal O_X)\) by Serre duality. We therefore simply refer to the genus of \(X\) and denote it \(g = H^1(X,\mathcal O_X)\).
For \(r = 2\) we have the degree-genus formula: \(g = \frac{(d-1)(d-2)}{2}\).
For \(r = 3\) it is no longer true that \(g\) depends solely on \(d\).
- Castelnnovo’s bound gives an upper bound to \(g\)
- A classification was given by Hartshorne.
For any \(r\) there is a generalization of Castelnnovo’s bound, but giving the possible values is an open problem.
Rank
Let’s instead ask a different question: given fixed \(g\) and \(d\), what are the possible values of \(r\)?
Riemann-Roch: \(r(L) - r(K_C - L) = d - g + 1\), which implies \(r(L) \geq \max \{-1, d - g + 1\}\). This gives a lower bound on \(r(L)\).
Clifford Theorem: \(r(L) \leq \frac{d}{2}\) if \(0 \leq d \leq 2g - 2\). This gives an upper bound on \(r(L)\).
When \(d \geq 2g - g\), the lower bound and upper bound of \(r\) coincide, and then \(d\) fully determines \(g\). For \(0\leq d\leq 2g - 2\) however, the upper and lower bounds for \(r\) do not coincide giving a “special region of line bundles”. This region is exactly what the Brill Noether theorem describes.
Theorem
(Brill Noether Theorem). If \(C\) is a general curve of genus \(g\) then there exists a line bundle \(g^d_r\) on \(C\) if and only if
\begin{align*} \rho(g,r,d) = g - (r + 1)(g - d + r) \geq 0. \end{align*}There’s an even stronger theorem:
Theorem
The locus of a line bundle \(g^r_d\) in \(Pic(X)\) has dimension \(\rho\) and is smooth and irreducible.