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# On the Image of Certain Extension Maps.~I

Published:2007-09-01
Printed: Sep 2007
• Israel Moreno Mejía
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## Abstract

Let $X$ be a smooth complex projective curve of genus $g\geq 1$. Let $\xi\in J^1(X)$ be a line bundle on $X$ of degree $1$. Let $W=\Ext^1(\xi^n,\xi^{-1})$ be the space of extensions of $\xi^n$ by $\xi^{-1}$. There is a rational map $D_{\xi}\colon G(n,W)\rightarrow SU_{X}(n+1)$, where $G(n,W)$ is the Grassmannian variety of $n$-linear subspaces of $W$ and $\SU_{X}(n+1)$ is the moduli space of rank $n+1$ semi-stable vector bundles on $X$ with trivial determinant. We prove that if $n=2$, then $D_{\xi}$ is everywhere defined and is injective.
 MSC Classifications: 14H60 - Vector bundles on curves and their moduli [See also 14D20, 14F05] 14F05 - Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 14D20 - Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}