Let be a genus curve over the field of complex numbers. I’ve been trying to understand a little about *special linear series* on : that is, low degree maps , or equivalently divisors on that move in a pencil. Once the degree is at least , any divisor will produce a map to (in fact, many maps), and these fit into nice families. In degrees , maps are harder to write down, and the families they form (for fixed ) 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 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 curve has a map to of degree at most , but for degrees below that, the “general” genus 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. *

**1. Hyperelliptic curves**

For , we are considering **hyperelliptic curves**: that is, curves with a degree two map to . An application of the Riemann-Hurwitz formula shows that the hyperelliptic map

must be branched over exactly points. In other words, it is a two-sheeted cover of , and the sheets come together at points. Since it is a degree two cover, it is necessarily Galois, and has a **hyperelliptic involution** over with those branch points as its fixed points, such that

When , is an elliptic curve (once one chooses an origin on ), and the hyperelliptic involution can be realized as with respect to the group law on . The resulting map can be realized, for a Weierstrass curve , by the function which respects the involution : in analytic terms, the hyperelliptic map is given by the Weierstrass -function. The ramification points of this map are the fixed points of the involution , or the four 2-torsion points on .

In this case, it’s important that the hyperelliptic map is not unique (even up to automorphisms of ). Namely, the hyperelliptic map depended upon the choice of an origin , and then was given by the divisor —which moved in a “pencil” and thus defined a map to . (In other words, the line bundle corresponding to the divisor had a two-dimensional space of sections .) However, a different would have provided a different line bundle and a different map to , except for three exceptional choices of .

However, in genus , the hyperelliptic map on a curve (if it exists) is unique. To see this, we use the following lemma, called the **basepoint-free pencil trick.**

Lemma 1Let be line bundles on a curve . Let be sections of without common zeros and consider the map

Then the kernel of this map is .

*Proof:* Indeed, one has an exact sequence of sheaves

where the first map sends a section of to the pair . This is precisely a Koszul-type complex, for the regular sequence —regularity follows because the vanishing loci are disjoint. Taking global sections gives the desired claim.

Let’s now suppose that is a hyperelliptic curve of genus and are two degree two line bundles with ; that means they’re generated by their global sections (since if they had a basepoint, one would get a degree one line bundle with sections). The basepoint-free pencil trick now shows that, if , then

where has degree four. Since , we must have

since for any line bundle on a curve of genus , we have for : otherwise we could keep subtracting points of to get a degree one map to .

Now let’s apply the basepoint-free pencil trick to and . We get

and, once again, equality holds since . Inductively, we get

Taking , we know that this has to be equal to , so that .

In other words, the choice of a hyperelliptic map is a *condition*, not extra data (modulo automorphisms of ). For example, when , the hyperelliptic map can be described as the *canonical* map: the map associated to the canonical line bundle. To specify a hyperelliptic curve of genus is thus equivalent to specifying points on over which the degree two cover is branched, modulo automorphisms of . In other words, the **moduli space of hyperelliptic curves** is given by

where is the *configuration space* of distinct (unordered) points of . In particular, the dimension is given by

so that hyperelliptic curves form a rather small subspace of the moduli space of curves, which has dimension . Moreover, it is a *unirational* variety: it admits a dominant rational map from a projective space. This is in sharp contrast to the moduli space , which is of general type for by a celebrated theorem of Harris and Mumford.

**2. Trigonal curves**

Let’s consider the next case: that of a degree three map (or a **trigonal** curve). We will also assume that the genus of is at least .

In other words, there exists a basepoint-free line bundle on of degree three, defining the map . In fact, in this case, we have

and so the map is associated to a *complete linear system.* One way to see this is to appeal to **Clifford’s theorem**, which states that:

Theorem 2 (Clifford)For a line bundle of degree at most on a curve , one has

Observe that is the dimension of the complete linear system associated to , i.e. the projective space . In this case, Clifford’s theorem shows that the linear system associated to a degree three line bundle on (if ) has dimension at most , which was our claim.

Every curve of genus is either hyperelliptic or trigonal. Given a genus two curve, we already know that it is hyperelliptic via the canonical map. Let’s look at the next two cases.

**Example 1** Given a genus three curve , if it is not hyperelliptic, the canonical map imbeds as a smooth plane quartic in , and projection from a point on is a degree three map from to .

**Example 2** Given a genus four curve, if it is not hyperelliptic, the canonical map imbeds as the (degree six) complete intersection of a quadric and a cubic in . Let’s admit this, and see how to produce the degree three map .

We need to produce three points on which move in a pencil. The Riemann-Roch theorem, in its “geometric” form, states that this is equivalent to finding such that the images of via the canonical imbedding live inside a line, a . In other words, impose one less than the expected number of linear conditions on differential 1-forms in : we have

rather than .

To produce these three points, observe that a quadric always contains a : in fact, lots of copies of . Now take the three points of intersection between and given by Bezout’s theorem; they are in and live on a line, and so move in a pencil. (This is the sort of argument that Brill-Noether theory does very efficiently, in higher genera when one doesn’t have as clear a picture of curves.)

Once again, trigonal divisors on a curve — degree three divisors that move in a pencil without base points — are *very* special divisors for large , and we should expect them to be in short supply.

Proposition 3A curve of genus cannot be both hyperelliptic and trigonal.

*Proof:* To see this, suppose given a degree two map and a degree three map . Equivalently, suppose given line bundles on of degrees with .

The basepoint-free pencil trick now implies that

If the genus is at least four, then is a degree five special divisor (of degree at most ) whose contradicts Clifford’s theorem.

If the genus is four, then the result fails. Namely, we can take a smooth two-dimensional quadric surface (i.e., a ), and take a smooth divisor of type . Given a smooth curve in of type in , the genus is given by , so if the genus is four. Such a curve comes with two natural degree three maps to , which must be distinct since the curve is imbedded in . In fact, it follows from this — since every (**edit: **not quite, some of these live on singular quadrics and are trigonal in only one way) nonhyperelliptic genus four curve is given by a -curve in — that the *general* curve of genus four has (at least) two distinct maps .

Proposition 4A curve of genus cannot be trigonal in two different ways.

*Proof:* Similarly, suppose there exist two different line bundles of degree and with . In this case, we can use the basepoint-free pencil trick (again!) to get

and that contradicts the equality case of Clifford’s theorem: is special and has degree too small to be the canonical divisor.

It’s interesting that this pattern does not persist: a curve (of high genus) can be tetragonal in infinitely many ways. To construct such examples, consider **bi-elliptic curves**: that is, curves with a degree two map for an elliptic curve. By increasing the branching, we can make of genus as high as we want. Then there are lots of degree four maps

given by using the (many distinct) degree two maps .

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