Real theta-characteristics on real projective curves (Q1866422)

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.