Gonality and genus of canonical components of character varieties

Autor: Alan W. Reid, Kathleen L. Petersen
Rok vydání: 2020
Předmět:
Zdroj: Characters in Low-Dimensional Topology. :263-292
ISSN: 1098-3627
0271-4132
DOI: 10.1090/conm/760/15295
Popis: Throughout the paper,M will always denote a complete, orientable finite volume hyperbolic 3-manifold with cusps. By abuse of notation we will denote by ∂M to be the boundary of the compact manifold obtained from M by truncating the cusps. Given such a manifold, the SL2(C) character variety of M , X(M), is a complex algebraic set associated to representations of π1(M)→ SL2(C) (see §4 for more details). Work of Thurston showed that any irreducible component of such a variety containing the character of a discrete faithful representation has complex dimension equal to the number of cusps of M . Such components are called canonical components and are denoted X0(M). Character varieties have been fundamental tools in studying the topology of M (we refer the reader to [23] for more), and canonical components carry a wealth of topological information aboutM , including containing subvarieties associated to Dehn fillings of M . WhenM has exactly one cusp, any canonical component is a complex curve. The aim of this paper is to study how some of the natural invariants of these complex curves correspond to the underlying manifold M . In particular, we concentrate on how the gonality of these curves behaves in families of Dehn fillings on 2-cusped hyperbolic manifolds. More precisely, we study families of 1-cusped 3-manifolds which are obtained by Dehn filling of a single cusp of a fixed 2-cusped hyperbolic 3-manifold, M . We write M(−, r) to denote the manifold obtained by r = p/q filling of the second cusp of M . To state our results we introduce the following notation. If X is a complex curve, we write γ(X) to denote the gonality of X, g(X) to be the (geometric) genus of X and d(X) to be the degree (of the specified embedding) of X. The gonality of a curve is the lowest degree of a map from that curve to C. Unlike genus, gonality is not a topological invariant of curves, but rather is intimately connected to the geometry of the curve. There are connections between gonality and genus, most notably the Brill-Noether theorem which gives an upper bound for gonality in terms of genus (see § 7) but in some sense these are orthogonal invariants. For example, all hyperelliptic curves all have gonality two, but can have arbitrarily high genus. Moreover, for g > 2, there are curves of genus g of different gonality. We refer the reader to § 3 for precise definitions. Our first theorem is the following.
Databáze: OpenAIRE
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