Reference request: base point freeness of $2\Theta$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T04:56:53Z http://mathoverflow.net/feeds/question/114790 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114790/reference-request-base-point-freeness-of-2-theta Reference request: base point freeness of $2\Theta$ jsm 2012-11-28T17:34:48Z 2012-11-28T19:05:23Z <p>Let $J$ be a Jacobian variety defined over a field $k$ and let $\Theta$ be a symmetric theta-divisor on $J$. </p> <p>It's shown (for instance) in the book Complex Abelian Varieties by Lange and Birkenhake that the linear system of $2\Theta$ is base point free if $k=\mathbb{C}$ and that it gives an embedding of the Kummer variety of $J$ into $\mathbb{P}^{2^g-1}$, where $g$ is the dimension of $J$.</p> <p>I'm looking for a reference that this holds for any algebraically closed field $k$. In fact $\mathrm{char}(k)\ne 2$ would suffice.</p> http://mathoverflow.net/questions/114790/reference-request-base-point-freeness-of-2-theta/114804#114804 Answer by Piotr Achinger for Reference request: base point freeness of $2\Theta$ Piotr Achinger 2012-11-28T19:05:23Z 2012-11-28T19:05:23Z <p>If $D$ is an ample divisor on an abelian variety, then $2D$ is base point free and $3D$ is very ample. One reference for this is Mumford's book "Abelian Varieties", II 6 and III 17.</p>