On theta characteristics of real algebraic curves
A real algebraic curve is a pair \((X,g)\) where \(X\) is a compact Riemann surface and \(g:X\to X\) is an antiholomorphic involution. Clearly, \(g\) acts on the set \(\Sigma\) of theta characteristics of \(X\). Moreover for \(\varepsilon=0,1\), let \(\Sigma_\varepsilon\) be the set of those \(M\in\Sigma\) such that the dimension of \(H^0(X, {\mathcal O}(M))\) is congruent to \(\varepsilon\pmod 2\). It is easily seen that \(g(\Sigma_\varepsilon) = \Sigma_\varepsilon\). There is a classical formula that provides the number of points in \(\Sigma_\varepsilon\) in terms of the genus \(p>1\) of \(X\). In the paper under review the author obtains the real counterpart of this. More precisely, if \(\Sigma_\varepsilon (\mathbb{R})\) and \(X(\mathbb{R})\) are (respectively), the set of points in \(\Sigma_\varepsilon\) (resp. in \(X)\) fixed by \(g\), the author computes the number of points in \(\Sigma_\varepsilon (R)\) in terms of \(p\) and the number \(s\) of connected components of \(X(\mathbb{R})\). As a nice example the author calculates the number of real bitangents to a non-singular plane curve of degree 4 whose real part has the maximum number of connected components. The paper is very short and essentially self-contained and well written.
scientific article; zbMATH DE number 1294973