Counting Bitangents with Stable Maps

Efficiently finding bitangents

We propose a new algorithm for the detection of bitangents in planar curves. It is much faster than an algorithm used by Rothwell et al. (95). Careful comparisons of the stability of detection using artificially transformed curves show that the new algorithm is as stable as the one proposed by Rothwell et al. An example for the use of the algorithm for object search is given.

Maps, Sheaves, and K3 Surfaces

The conjectural equivalence of curve counting on Calabi-Yau 3-folds via stable maps and stable pairs is discussed. By considering Calabi-Yau 3-folds with K3 fibrations, the correspondence naturally connects curve and sheaf counting on K3 surfaces. New results and conjectures (with D. Maulik) about descendent integration on K3 surfaces are announced. The recent proof of the Yau-Zaslow conjecture...

Lifting Tropical Bitangents

We study lifts of tropical bitangents to the tropicalization of a given complex algebraic curve together with their lifting multiplicities. Using this characterization, we show that generically all the seven bitangents of a smooth tropical plane quartic lift in sets of four to algebraic bitangents. We do this constructively, i.e. we give solutions for the initial terms of the coefficients of th...

Computing bitangents for ellipses

We show how to compute the bitangents of two ellipses, and how to evaluate the predicates required to compute objects like convex hulls or visibility complexes for ellipses. Also, we show how to compute the tangency points of the bitangents.

Relative Compactness and Product Stable Quotient Maps