Distortion maps for genus two curves

A pr 2 00 9 Abel maps for curves of compact type

Recently, the first Abel map for a stable curve of genus g ≥ 2 has been constructed. Fix an integer d ≥ 1 and let C be a stable curve of compact type of genus g ≥ 2. We construct two d-th Abel maps for C, having different targets, and we compare the fibers of the two maps. As an application, we get a characterization of hyperelliptic stable curves of compact type with two components via the 2-n...

Landmark constrained genus-one surface Teichmüller map applied to surface registration in medical imaging

We address the registration problem of genus-one surfaces (such as vertebrae bones) with prescribed landmark constraints. The high-genus topology of the surfaces makes it challenging to obtain a unique and bijective surface mapping that matches landmarks consistently. This work proposes to tackle this registration problem using a special class of quasi-conformal maps called Teichmüller maps (T-...

Invariant Curves for Birational Surface Maps

We classify invariant curves for birational surface maps that are expanding on cohomology. When the expansion is exponential, the arithmetic genus of an invariant curve is at most one. This implies severe constraints on both the type and number of irreducible components of the curve. In the case of an invariant curve with genus equal to one, we show that there is an associated invariant meromor...

The Arithmetic and the Geometry of Kobayashi Hyperbolicity

(1) The dimension of the pluricanonical series, h(C,mK), grows linearly with m for curves of genus at least two. (2) The canonical/cotangent bundle of a curve of genus at least two is ample. (3) A curve of genus at least two admits a hyperbolic metric with constant negative curvature. (4) Curves of genus at least two are uniformized by the unit disc, hence they do not admit any non-constant hol...

Efficiently Computable Distortion Maps for Supersingular Curves