DC Field | Value | Language |
---|---|---|
dc.contributor.advisor | Lee, Yongnam | - |
dc.contributor.advisor | 이용남 | - |
dc.contributor.author | Kim, Jeong-Seop | - |
dc.date.accessioned | 2023-06-22T19:33:50Z | - |
dc.date.available | 2023-06-22T19:33:50Z | - |
dc.date.issued | 2022 | - |
dc.identifier.uri | http://library.kaist.ac.kr/search/detail/view.do?bibCtrlNo=1007822&flag=dissertation | en_US |
dc.identifier.uri | http://hdl.handle.net/10203/308564 | - |
dc.description | 학위논문(박사) - 한국과학기술원 : 수리과학과, 2022.8,[i, 38 p. :] | - |
dc.description.abstract | When a vector bundle $E$ on an algebraic curve is stable, every symmetric power $S^k E$ is known to be stable for sufficiently general $E$. This thesis deals with the question of which $E$ has not stable $S^k E$ in the case where $E$ has rank $2$. As an answer to the question, it is shown that if $S^k E$ is not stable, then $S^m E$ is not stable for some $m=2$, $3$, $4$ or $6$, and moreover, it is shown that if $S^k E$ is not stable, then $S^l E$ is destabilized by a line subbundle for some $l\geq k$. So the stability of $S^k E$ can be rephrased by the existence of a curve with zero self-intersection number on the ruled surface $\mathbb{P}_C(E)$ associated to $E$, and as its corollary, it is shown that every symmetric power $S^k E$ is stable for general $E$. That is, when $E$ has rank $2$, it is possible to remove the 'sufficiently' assumption in the known result. Also, this thesis treats a classification of $E$ in the case of $k=2$ and $3$. When $k=2$, as $E$ with $S^2 E$ being not stable is a vector bundle with orthogonal structure, a relation between known descriptions are investigated, and when $k=3$, it is shown that there exists $E$ with stable $S^2 E$ but not stable $S^3 E$ (when the genus of curve is larger than or equal to $2$). Lastly, this thesis shows that if $S^2 E$ is stable but $S^k E$ is not stable for some $k>2$, then $E$ with trivial determinant is trivialized over an unramified finite covering of given curve. This thesis is based on a paper by the author published in International Journal of Mathematics, and it generalizes the result by removing the assumption on the degree of $E$. | - |
dc.language | eng | - |
dc.publisher | 한국과학기술원 | - |
dc.subject | vector bundle▼asymmetric power▼astability▼aruled surface▼acone of curves | - |
dc.subject | 벡터 번들▼a대칭 거듭제곱▼a안정성▼a선직면▼a곡선 원추 | - |
dc.title | Stability of symmetric powers of vector bundles of rank two on a curve | - |
dc.title.alternative | 대수 곡선 상의 차수가 2인 벡터 번들의 대칭 거듭제곱의 안정성 | - |
dc.type | Thesis(Ph.D) | - |
dc.identifier.CNRN | 325007 | - |
dc.description.department | 한국과학기술원 :수리과학과, | - |
dc.contributor.alternativeauthor | 김정섭 | - |
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