On derived categories of algebraic surfaces constructed via Q-Gorenstein smoothings = 유리-고렌스타인 매끄러움을 통해 건설되는 대수 곡면의 유도 범주에 대하여

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This thesis mainly consists of the study of line bundles on algebraic surfaces constructed via $\BbbQ$ -Gorenstein smoothing. Recently, $\BbbQ$ -Gorenstein smoothing is used to construct new algebraic surfaces, however, it is relatively unknown how the geometric information varies along the smoothing. Hacking's weighted blow up technique, which has been used to construct exceptional vector bundles on a surface from its singular degenerations, is applied to study the line bundles on the surfaces obtained by $\BbbQ$ -Gorenstein smoothing. The main idea is to lift information about divisors to a (possibly non-Cartier) divisors on the central fiber of the smoothing. It turns out that numerical invariants of divisors, such as holomorphic Euler characteristics, can be understood via information from the central fiber. Also, the cohomological properties of the divisors can be studied with the same method, even if it is less efficient in general. In some nice and simple examples of $\BbbQ$ -Gorenstein smoothings, cohomologies of divisors are efficiently controlled under this method. As an example, cohomological properties of divisors in Dolgachev surfaces of type (2,3) will be studied in details. This leads to the presentation of semiorthogonal decompositions of the derived category into exceptional collection of 12 line bundles and a phantom category. Using the same method, we study the nonminimal Enriques surfaces, and establish semiorthogonal decompositions of the derived categories into 13 line bundles and quasi-phantom categories.
Advisors
Lee, Yongnamresearcher이용남researcher
Description
한국과학기술원 :수리과학과,
Publisher
한국과학기술원
Issue Date
2017
Identifier
325007
Language
eng
Description

학위논문(박사) - 한국과학기술원 : 수리과학과, 2017.2,[i, 69 p. :]

Keywords

Q-Gorenstein smoothing; elliptic surfaces; derived categories; semiorthogonal decompositions; (quasi-)phantom categories; 유도 범주; 유리-고렌스타인 매끄러움; 준직교분해; 타원 곡면; (준)환영 범주

URI
http://hdl.handle.net/10203/241904
Link
http://library.kaist.ac.kr/search/detail/view.do?bibCtrlNo=675757&flag=dissertation
Appears in Collection
MA-Theses_Ph.D.(박사논문)
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