Gauss Bonnet Theorem for Singular Curves in CPn
| dc.contributor.author | Levy, Benjamin Philip | en |
| dc.contributor.committeechair | Haskell, Peter E. | en |
| dc.contributor.committeemember | Douglas, Daniel Charles | en |
| dc.contributor.committeemember | Yang, Yun | en |
| dc.contributor.department | Mathematics | en |
| dc.date.accessioned | 2026-07-15T08:01:15Z | en |
| dc.date.available | 2026-07-15T08:01:15Z | en |
| dc.date.issued | 2026-07-14 | en |
| dc.description.abstract | This paper provides a generalization of the Gauss-Bonnet theorem to singular irreducible projective algebraic curves. For any singular curve $C$ with singularity set $Sigma$, we prove that if one integrates the Gaussian curvature provided by restricting the Fubini-Study metric to the noncompact manifold $C setminus Sigma$ that the result is well defined and equal $2pi$ times the Euler characteristic of $C$'s desingularization plus finitely many correction terms specified by the structure of $C$'s singularities. | en |
| dc.description.abstractgeneral | A surface is a geometric object that is locally Euclidean in the sense that, at a small scale, the surface can be approximated well, for many purposes, by a piece of paper. Examples include a sphere (which models a basketball, ignoring the air inside) and a torus (which models the crust of a doughnut, ignoring the cooked dough inside). Physical applications include electric and gravitational equipotential surfaces. The locally Euclidean structure of a surface allows calculus to be used to calculate the surface's Gaussian curvature, a measure of the rate at which the direction the surface is facing changes as you move from point to point on the surface. A surface's combinatorial topology is revealed by subdividing the surface into polygonal regions, as soccer balls have traditionally been constructed. The Gauss-Bonnet theorem says that the two perspectives are related: the integral of Gaussian curvature over the surface equals $2pi$ times the surface's Euler characteristic (number of faces minus number of edges plus number of vertices in the polygonal decomposition). This thesis answers the question of how these quantities are related when a surface has singularities. Surfaces that are the solution sets of polynomial equations in two complex variables are the first examples studied in algebraic geometry. Some of these solution sets are known as singular surfaces, which are not true surfaces because they have isolated singularities, points where the set is not locally Euclidean. Gaussian curvature is not defined at these points. Every singular surface has a desingularization, a surface without singularities that maps to the singular surface in a way that makes the desingularization the best nonsingular approximation of the singular surface. This thesis shows that the integral of Gaussian curvature over the nonsingular points of a surface with singularities is finite, with value equal to the Euler characteristic of the singular surface's desingularization plus correction terms calculated using the map from the desingularization to the singular surface. | en |
| dc.description.degree | Master of Science | en |
| dc.format.medium | ETD | en |
| dc.identifier.other | vt_gsexam:47444 | en |
| dc.identifier.uri | https://hdl.handle.net/10919/143658 | en |
| dc.language.iso | en | en |
| dc.publisher | Virginia Tech | en |
| dc.rights | Creative Commons Attribution-ShareAlike 4.0 International | en |
| dc.rights.uri | http://creativecommons.org/licenses/by-sa/4.0/ | en |
| dc.subject | Gauss-Bonnet | en |
| dc.subject | Singular Curves | en |
| dc.title | Gauss Bonnet Theorem for Singular Curves in CPn | en |
| dc.type | Thesis | en |
| thesis.degree.discipline | Mathematics | en |
| thesis.degree.grantor | Virginia Polytechnic Institute and State University | en |
| thesis.degree.level | masters | en |
| thesis.degree.name | Master of Science | en |
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