A Proof of Looijenga's Conjecture via IntegralAffine Geometry
 Title:
 A Proof of Looijenga's Conjecture via IntegralAffine Geometry
 Author(s):
 Engel, Philip
 Thesis Advisor(s):
 Friedman, Robert
 Date:
 2015
 Type:
 Dissertations
 Department(s):
 Mathematics
 Persistent URL:
 http://dx.doi.org/10.7916/D8028QGQ
 Notes:
 Ph.D., Columbia University.
 Abstract:
 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 integralaffine surfaces, twodimensional manifolds whose transition functions are valued in the integralaffine 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.
 Subject(s):
 Mathematics
 Item views
 218
 Metadata:

text  xml
 Suggested Citation:
 Philip Engel, 2015, A Proof of Looijenga's Conjecture via IntegralAffine Geometry, Columbia University Academic Commons, http://dx.doi.org/10.7916/D8028QGQ.