A short post on bitangents

Today’s post will be, as the title says, a bit short.  It will, more-or-less finish our current discussion of theta characteristics, and then we’ll get back to something else.  But we’ll derive a nice case of a classical formula.

Fix a curve $C$.  A vanishing thetanull on the curve is a theta characteristic $L$ with $h^0(C,L)>1$.  This occurs if an even theta characteristic is effective, or if an odd theta characteristic gives a map to the plane.  This is actually a rare occurrence, and a generic curve doesn’t have a vanishing thetanull.  So now we’re restricting to $C$ generic.

A theta characteristic on such a curve is effective if and only if it is odd.  On top of that, as $L^2\cong \omega_C$, the divisor $D$ associated to $L$ can be doubled to give a canonical divisor.  What does this mean geometrically? It means that if we can the canonical embedding of the curve, then $2D$ is a hyperplane section.

Now for the nice classical consequence: let $g=3$.  Then the canonical curve lies in $\mathbb{P}^2$, and is degree 4.  So we have a smooth plane quartic.  Generically, there are no vanishing thetanulls, so every divisor $D$ such that $2D$ is the intersection of $C$ with a line comes from a unique odd theta characteristic, and is uniquely determined by it.  So $D$ is a degree 2 divisor, so the line intersects the curve in two points.  As $C\cap \ell=2D$, that means that the line is tangent to the curve at both points.  Thus, for a generic curve of genus $3$ (that is, a generic quartic plane curve), the number of bitangents is exactly the number of odd theta characteristics.  Last time, we computed this number: $2^2(2^3-1)=4\times 7=28$.

This 28 has interesting connections to many things, but I’ll only mention one.  They are the zero locus of a quadric of Art invariant 1 on $\mathbb{F}_2^6$.  If we projectivize, so that we have a quadric on $\mathbb{P}^5(\mathbb{F}_2)$, then the locus consists of 27 points, and if we say that two points are incident if they lie on an isotropic line, then we get the incidence relation of lines on a smooth cubic surface! It’s even better than that.  If we take a smooth cubic surface in $\mathbb{P}^3$, and project to the plane from a point not on any line, then we get a double cover of the plane branched at a smooth quartic.  Then, the 27 lines go to 27 of the bitangents, and the other one is the exceptional locus of the blowup! So, odd theta characteristics correspond to bitangents, and for quartic curves, these correspond to the lines on a cubic surface, plus a point not on any of the lines.