|
3:30 pm, Seminar Hall Semi-Stability of Certain Vector Bundles on Elliptic Curves Amit Kumar Singh Institute of Mathematical Sciences, Chennai. 01-10-19 Abstract Let $L$ be a line bundle of degree $d$ on an elliptic curve $C$ and $\varphi : C\rightarrow \mathbb{P}^{n}$ is a morphism given by a sub-linear system of the complete linear system $\vert L \vert$ of dimension $n+1$. When $d$ = 4, $n$ = 2, we prove that $\varphi^{*}T_{\mathbb{P}^{n}}$ is semi-stable if $\text{deg}(\varphi(C))>1$. Moreover, we prove that $\varphi^{*}T_{\mathbb{P}^{n}}$ is isomorphic to direct sum of two isomorphic line bundles if and only if $ \deg (\varphi(C))=2$. Conversely, for any rank two semi-stable vector bundle $E$ on an elliptic curve $C$ of degree 4, there is a non-degenerate morphism $\varphi : C \rightarrow \mathbb{P}^n$ such that $\varphi^* T_{\mathbb{P}^n}(-1)= E$. More precisely, $E$ is isomorphic to direct sum of two isomorphic line bundles if and only if $\deg(\varphi(C))=2$. Further $E$ is either indecomposable or direct sum of non-isomorphic line bundles if and only if $\deg(\varphi(C))=4$. When $d$ = 5, $n$ = 3, we compute the Harder-Narasimhan filtration of $\varphi^*T_{\mathbb{P}^n}$.
|
