Type
Text
Type
Dissertation
Advisor
Sullivan, Dennis Parnell | Zinger, Aleksey | McDuff, Dusa | Starr, Jason Michael.
Date
2011-12-01
Keywords
bundle, enumerative, Euler, geometry | Mathematics
Department
Department of Mathematics
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/71492
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
Enumerative geometry is a rich and fascinating subject that has been studied extensively by algebraic geometers. In our thesis however, we approach this subject using methods from differential topology. The method comprises of two parts. The first part involves computing the Euler class of a vector bundle and evaluating it on the fundamental class of a manifold. This is straightforward. The second part involves perturbing a section and computing its contribution near the boundary. This is usually difficult. We have used this method to compute how many degree $d$ curves are there in $\C\P^2$ that pass through $\frac{d(d+3)}{2} -(\delta + m)$ points having $\delta$ nodes and one singularity of codimension $m$ provided $\delta+m\leq 7$. We also indicate how to extend this approach if $\delta+m$ is greater than $7$. | 102 pages
Recommended Citation
Mukherjee, Ritwik, "Enumerative Geometry via Topological Computations" (2011). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 698.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/698