3
$\begingroup$

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...

Improve this question
$\endgroup$

1 Answer 1

Reset to default
10
$\begingroup$

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 $X.$

Improve this answer
$\endgroup$
1
  • $\begingroup$ 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? $\endgroup$
    – IMeasy
    Commented Oct 22, 2012 at 15:37

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .