Let ${C}$ be a genus ${g}$ curve over the field ${\mathbb{C}}$ of complex numbers. I’ve been trying to understand a little about special linear series on ${C}$: that is, low degree maps ${C \rightarrow \mathbb{P}^1}$, or equivalently divisors on ${C}$ that move in a pencil. Once the degree is at least ${2g + 1}$, any divisor will produce a map to ${\mathbb{P}^1}$ (in fact, many maps), and these fit into nice families. In degrees ${\leq 2g-2}$, maps ${C \rightarrow \mathbb{P}^1}$ are harder to write down, and the families they form (for fixed $C$) aren’t quite as nice.

However, it turns out that there are varieties of special linear series—that is, varieties parametrizing line bundles of degree ${\leq 2g-2}$ with a certain number of sections, and techniques from deformation theory and intersection theory can be used to bound below and predict their dimensions (the predictions will turn out to be accurate for a general curve). For instance, one can show that any genus ${g}$ curve has a map to ${\mathbb{P}^1}$ of degree at most ${\sim \frac{g}{2}}$, but for degrees below that, the “general” genus ${g}$ curve does not admit such a map. This is the subject of the Brill-Noether theory.

In this post, I’d just like to do a couple of low-degree examples, to warm up for more general results. Most of this material is from Arbarello-Cornalba-Griffiths-Harris’s book Geometry of algebraic curves.  (more…)