A Proof of Looijenga's Conjecture via Integral-Affine Geometry
A cusp singularity is a surface singularity whose minimal resolution is a reduced cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In 1981, Looijenga proved that whenever a cusp singularity is smoothable, the minimal resolution of the dual cusp is an anticanonical divisor of some smooth rational surface. He conjectured the converse. This dissertation provides a proof of Looijenga's conjecture based on a combinatorial criterion for smoothability given by Friedman and Miranda in 1983, and explores the geometry of the space of smoothings. The key tool in the proof is the use of integral-affine surfaces, two-dimensional manifolds whose transition functions are valued in the integral-affine transformation group. Motivated by the proof and recent work in mirror symmetry, we make a conjecture regarding the structure of the smoothing components of a cusp singularity.
Academic Commons
Engel, Philip
Author
Friedman, Robert
Thesis advisor
Columbia University. Mathematics
Originator
Theses
English
Mathematics
text
2015
eng
2017-06-12T17:37:17Z
2017-06-12T18:36:39Z
Ph.D.
2
Mathematics
Columbia University
10.7916/D8028QGQ