So, I don’t really have a full length post in me at the moment. However, here’s a nice trick that I’ve learned recently, plus some motivation. With this, it’s easy to write down explicitly a curve of arbitrary degree in the plane, which has specified nodes. (So long as it’s possible to have that many nodes.)
First up, the motivation for why we want to write down lots of plane curves with nodes. First, up, we quote a theorem, which is proved thoroughly anywhere that curves are seriously investigated:
Theorem: Every algebraic curve can be embedded in .
A route to proving this is to show that hyperelliptic curves are plane curves, and that nonhyperelliptic curves are embedded by the canonical embedding, and that taking projections of curves gives isomorphisms down to .
However, not every curve is actually isomorphic to a plane curve. But the second theorem we’re going to quote gives us a way to represent every curve in the plane:
Theorem: Let be a curve in . Then there exists a point such that projection from the point maps surjectively onto a plane curve with only nodes as singularities.
So every curve can be represented as a curve of degree with nodes for some satisfying . So it’d be nice to have a good way to write down plane curves of a given degree and number of nodes, which then determines the genus.
The engine that makes this all work is Bertini’s Theorem. We’re going to construct a linear system of curves with some base points, and then a generic one will be the curve we want. In fact, it will be a one dimensional linear system, which we’ll refer to as a pencil.
So, fix a degree and the number of nodes that you want , so long as is small enough that such a curve can exist. In fact, we’re going to do this by constructing two curves with the maximal possible number of nodes.
Fix points in the plane, and take two configurations of lines such that the points are intersections of lines in each configuration, and no other points are. Also require that they have no whole lines in common. This is fairly easy to do by hand, because we’re just taking unions of lines. These give polynomials and .
Using and , we define a pencil of curves by looking at the zero locus of where are homogeneous coordinates on . Bertini then tells us that a general element of this pencil is smooth away from the points, which we made into base points (and which are the only ones). So the singular locus of the general curve here consists of points.
Finally, moving in the pencil will preserve the singularity types of these base points, which were originally nodes. Thus, we have constructed a whole family of genus , degree plane curves with nodes, so we can get concrete views of any curve in this way.