most recent 30 from http://mathoverflow.net 2013-05-26T05:20:01Z http://mathoverflow.net/feeds/question/110338 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110338/octic-k3s-inside-cubic-4-folds IMeasy 2012-10-22T15:00:10Z 2012-10-22T15:31:43Z

<p>From the Thesis of B.Hassett I seem to understand that a smooth cubic 4-fold $X$ containing a $\mathbb{P}^2$ should contain also a octic K3, but I cannot see a natural way by which this K3 octic could appear inside $X$. How do you see that?</p> <p>One wild guess is: via linkage of a quartic 3-fold that contains a quadric surface contained in $X$... but this doesn't seem very consistent...</p>

http://mathoverflow.net/questions/110338/octic-k3s-inside-cubic-4-folds/110341#110341 Yusuf Mustopa 2012-10-22T15:31:43Z 2012-10-22T15:31:43Z

<p>The $\mathbb{P}^{2}$ contained in your cubic fourfold $X$ is cut out by linear forms (say) $L_{1},L_{2},$ and $L_{3}.$ Since the homogeneous ideal of $X$ is contained in the homogeneous ideal generated by $L_{1},L_{2}$ and $L_{3},$ there exist quadrics $Q_{1},Q_{2},$ and $Q_{3}$ such that $X={L_{1}Q_{1}+L_{2}Q_{2}+L_{3}Q_{3}=0}.$ The octic K3 cut out by $Q_{1},Q_{2}$ and $Q_{3}$ is easily seen to lie in $X.$</p>