Half-canonical series on algebraic curves

Abstract

Denote by Jf the locus in the moduli space of curves of genus g of those curves which have a theta-characteristic of (projective) dimension at least r. We give an upper bound for the dimension of Jtr and we determine this dimension completely for r ^ 4. For r < 4, we prove also that a generic point in every component of J(r has a single theta-characteristic of this dimension. 0. Introduction. Let Jt g be the moduli space of smooth, complete curves of genus g over the complex field C. We investigate the subloci Jlrg of J(' These are defined as the loci of curves having a theta-characteristic (i.e. a line bundle L such that L ® L = Kc) of (projective) dimension at least r and of the same parity as r. By Clifford's Theorem, it is clear that J(r is empty if r > \(g-1). On the other hand, one can easily see that hyperelliptic curves have theta-characteristics of all dimensions r with 0 < r < \~(g-1). In [H], Harris proves (0.1) Theorem (cf. [H, Theorem (1.10)]). Let X-> S be a family of curves and La line bundle on X such that the restriction L(s) to every fiber X(s) satisfies L2(s)-Kx(s)-Then the subset of S {s G S\k°(X(s),L(s))> r+ landh°(X(s),L(s)

Pencils on real curves

“…Now assume \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\rho ^1_k(g)\ge 0$\end{document}. From the arguments in 24, Appendix] it follows that each component of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H_k(\pi )(\mathbb {C})\ ($\end{document}or \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^R_k(\pi )(\mathbb {C}))$\end{document} dominates \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$S(\mathbb {C})$\end{document}. Let \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^{\prime }_k(\pi )(\mathbb {C})\ ($\end{document}resp.…”

Theta line bundles and the determinant of the Hodge bundle

“…We use the following result from [8], [9]. Let α :M g −→ M g denote the covering of even theta characteristics in J g−1 .…”

Moduli spaces of curves with effective 𝑟-spin structures

“…In fact, Teixidor i Bigas showed in [9] that this codimension is at least 2i − 1. It is plausible that similar analysis is applicable to more general loci S Convention.…”