## Real theta-characteristics on real projective curves.(English)Zbl 1056.14034

Let $$X$$ be a, not necessarily irreducible or nonsingular, reduced projective real algebraic curve, and $$X_{\mathbb{C}}$$ its complexification. Let $$\mathcal F$$ be a torsion-free sheaf on $$X_{\mathbb{C}}$$ that is of rank $$1$$ on each irreducible component of $$X_{\mathbb{C}}$$. The sheaf $$\mathcal F$$ is a theta-characteristic on $$X_{\mathbb{C}}$$ if it is isomorphic to the sheaf $${\mathcal Hom}({\mathcal F},\omega)$$, where $$\omega$$ is the dualizing sheaf on $$X_{\mathbb{C}}$$. The theta-characteristic $$\mathcal F$$ on $$X_{\mathbb{C}}$$ is said to be real, in this paper, if it is isomorphic to its complex conjugate.
The author proves the existence of a real theta-characteristic $$\mathcal F$$ on $$X_{\mathbb{C}}$$ that is completely singular and freely full. Completely singular means that the locus where $$\mathcal F$$ is not locally free coincides with the singular locus of $$X_{\mathbb{C}}$$. Freely full means that $$\mathcal F$$ is induced by an invertible sheaf through some proper birational morphism $$Y\rightarrow X_{\mathbb{C}}$$.
Similar statements are proved for even and odd theta-characteristics.

### MSC:

 14H20 Singularities of curves, local rings 14P99 Real algebraic and real-analytic geometry
Full Text: