Let ${C}$ be an algebraic curve over ${\mathbb{C}}$. A theta characteristic on ${C}$ is a (holomorphic or algebraic) square root of the canonical line bundle ${K_C}$, i.e. a line bundle ${L \in \mathrm{Pic}(C)}$ such that

$\displaystyle L^{\otimes 2} \simeq K_C.$

Since the degree of ${K_C}$ is even, such theta characteristics exist, and in fact form a torsor over the 2-torsion in the Jacobian ${J(C) = \mathrm{Pic}^0(C)}$, which is isomorphic to ${H^1(C; \mathbb{Z}/2\mathbb{Z}) \simeq (\mathbb{Z}/2\mathbb{Z})^{2g}}$.

One piece of geometric motivation for theta characteristics comes from the following observation: theta characteristics form an algebro-geometric approach to framings. By a theorem of Atiyah, holomorphic square roots of the canonical bundle on a compact complex manifold are equivalent to spin structures. In complex dimension one, a choice of a spin structure is equivalent to a framing of ${M}$. On a framed manifolds, there is a canonical choice of quadratic refinement on the middle-dimensional mod ${2}$ homology (with its intersection pairing), which gives an important invariant of the framed manifold known as the Kervaire invariant. (See for instance this post on the paper of Kervaire that introduced it.)

It turns out that the mod ${2}$ function ${L \mapsto \dim H^0(C, L)}$ on the theta characteristics is precisely this invariant. In other words, theta characteristics give a purely algebraic (valid in all characteristics, at least ${\neq 2}$) approach to the Kervaire invariant, for surfaces!

Most of the material in this post is from two papers: Atiyah’s Riemann surfaces and spin structures and Mumford’s Theta characteristics of an algebraic curve. (more…)