User franz - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:15:06Z http://mathoverflow.net/feeds/user/25682 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13990/evidences-on-hartshornes-conjecture-references/104620#104620 Answer by Franz for Evidences on Hartshorne's conjecture? References? Franz 2012-08-13T15:08:48Z 2012-08-14T10:52:42Z <p>I think the original statement of Hartshorne was that for smooth $X \subset \mathbb{P}^N$ with $3/2 \dim(X) > N$, then $X$ is a complete intersection (you recover your statement from this conjecture with the famous construction of Serre). Being a complete intersection implies that $H^0(\mathbb{P}^N, O_{\mathbb{P}^N}(k)) \rightarrow H^0(X,O_X(k))$ is surjective for all $k$. A variety with such a surjective map for a fixed $k$ is called $k$-normal.</p> <p>With surprisng new "topologico-geometric" methods, Fyodor Zak proved in the beginning of the 80's that if $3/2\dim(X) + 1 > N$ then $X$ is $1$-normal. He proved morever that for $3/2\dim(X) + 1 = N$ there exist only $4$ varieties which are not $1$-normal namely : the 2nd Veronese embedding of $\mathbb{P}^2$ projected down into $\mathbb{P}^4$, the Segre embedding of $\mathbb{P}^2 \times \mathbb{P}^2$ projected down into $\mathbb{P}^7$, the Plucker embedding of the grassmannian of $\mathbb{C}^2 \subset \mathbb{C}^6$ projected down into $\mathbb{P}^{13}$ and the closed orbit of $E_6$ in $\mathbb{P}^{26}$ projected down into $\mathbb{P}^{25}$. You can read his wonderful book "<strong>Tangents and Secants of Algebraic Varieties</strong>" to get complete proofs of these facts.</p> <p>If you restrict to the case of varieties defined by quadratic equations (which is a bit limitated but still interesting as a step toward a more complete answer), then Hartshorne conjecure has been proved in that setting by Ionescu and Russo (see <a href="http://arxiv.org/abs/0909.2763" rel="nofollow">http://arxiv.org/abs/0909.2763</a>). Note also that Hartshorne conjecture completely fails for other toric projective Fano varieties, see <a href="http://arxiv.org/abs/1005.5546" rel="nofollow">http://arxiv.org/abs/1005.5546</a>).</p> http://mathoverflow.net/questions/13990/evidences-on-hartshornes-conjecture-references/104620#104620 Comment by Franz Franz 2012-08-13T22:44:45Z 2012-08-13T22:44:45Z That's interesting, can you give the title of the paper? Nevertheless I think the result of the last reference I gave is a bit more subtle than the result by Klyachko you metionned. Indeed the bundles constructed by these authors are (if I remember correctly) never toric.