Singularities, sails and kites

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Thesis Type: Postgraduate

Institution Of The Thesis: Galatasaray Üniversitesi, Fen-Edebiyat Fakültesi, Matematik, Turkey

Approval Date: 2016

Thesis Language: English

Student: ERCAN BALCI

Supervisor: MERAL TOSUN

Open Archive Collection: AVESIS Open Access Collection

Abstract:

Plane curve singularities are easy to resolve and\hspace{0.07cm} we can associate two invariants to their desingularization process: Enriques trees, dual graphs. These invariants helps to read the geometry of each blowing up in the desingularization process. We define a new tool called multiple sails for curve singularities using their Enriques trees and we associate kites to multiple sails. We show that multiple sails contains both of the informations we can get from dual graphs or Enriques trees. In fact, multiple sails contains dual graphs and Enriques trees together as structure. The main aim of this thesis is to generalize the tools we have defined on curve singularities for the case of some special surface singularities especially for minimal singularities. For this purpose, we define royal sails and we use the notion depth which is defined for vertices of dual graphs associated to strong desingularizations. For minimal singularities, depth of a vertex is equal to number of necessary blow-ups to make corresponding irreducible component appear. At last, we try to generalize the same construction for the case of rational singularities.