Research Specifications

For a general $p$ on a smooth projective algebraic $C$ and for a very ample line bundle $\Gamma$ on $C$ we compare $V^r_d(\Gamma)$ and $V^r_d(\Gamma(-p))$, the varieties of secant divisors associated to the line bundles $\Gamma$ and $\Gamma(-p)$. As consequences of this comparision we obtain Mumford type classifications for $C$. The next main result is to obtain a symmetric behavior for dimension of $V^r_d(\Gamma)$. We do this by entering a suitable sub-line bundle $H$ of $\Gamma$ in to the game and studying $V^r_d(H)$ in comarision with $V^r_d(\Gamma)$. Meanwhile, a secondary result in our study asserts that $V^r_d(\Gamma)$ is a divisor in $V^r_d(\Gamma(-p))$.