Pencils on real curves

Abstract

We consider coverings of real algebraic curves to real rational algebraic curves. We show the existence of such coverings having prescribed topological degree on the real locus. From those existence results we prove some results on Brill‐Noether Theory for pencils on real curves. For coverings having topological degree \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\underline{0}$\end{document} we introduce the covering number k and we prove the existence of coverings of degree 4 with prescribed covering number.

Pencils on separating $$(M-2)$$ -curves

“…Hence the condition H 1 (X, L ⊗2 ) = 0 implies π −1 k (s 0 ) has dimension 2k−g +1 and it is smooth at [f ]. In [5] we introduced the topological degree of f . Choose an orientation on P 1 (R).…”

“…There exists a real algebraic deformation π : X → I with I a small neighborhood of 0 in [0, +∞[⊂ R such that π −1 (0) = Γ 0 and for t > 0 the curve X t = π −1 (t) is a smooth real complete curve of genus g such that X t (R) has n + 1 connected components (see e.g. [13,Section 7], it can be shown directly by using part of Construction II in [5]). We can assume for all t ∈ I the curve X t is not hyperelliptic.…”

“…Proof. From [5] we know there exists a dividing (M-2)-curve X such that sepgon(X) = g − 1. Assume X is of special type.…”

The separating gonality of a separating real curve

“…[3, Chapter V]). Using results concerning pencils on Riemann surfaces we obtain a proof of the theorem in case s = 1 (this argument resembles those used in [4,Section 3]). Then we take a suited separating real curve X ′ with s(X ′ ) = 1 and on X ′ C we identify closed points associated to some chosen non-real points on X ′ .…”

Real fibered morphisms and Ulrich sheaves

“…While in the context of hyperbolicity real fibered morphisms appear as linear projections there are also applications where this is not the case. For example, the existence of real fibered morphisms from curves to the projective line and their properties have been studied by several authors [17,18,26,35]. Typical questions include but are not limited to general existence results [26] or constructions of real fibered morphisms of small degree [17].…”