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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?

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...

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up vote 9 down vote accepted

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 $X.$

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Cool. What's the relation between these quadrics and the quadric 3-folds obtained by projecting off $\mathbb{P}^2$? Are the 3-fold quadrics the restriction of the the 4-fold ones? – IMeasy Oct 22 '12 at 15:37

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