I recently learnt that the hypersurface $$ S:=(x^2+y^3+z^{11}+w^{66}=0) \subset \mathbb{P}(33,22,6,1) $$ is birational to a K3 surface. This is surprising because the surface is quasi-smooth, well-formed, and its amplitude is $66-(33+22+11+1)=4$ which implies $$ \omega_S=\mathcal{O}_{S}(4) $$ This example is due to Dolgachev and Kondo and discussed further by Brunce and Schimmrigk [pg. 10, 1]. The issue is that $S$ contains a $-1$-curve supporting some cyclic quotient singularities. Then $S$ is not minimal.

My questions are:

1) A normal quasismooth surface in a weighted projective space has cyclic quotient singularities which are log terminal and rational. Therefore, they do not change the kodaira dimension of the surface. How can we determine the kodaira dimension of $S$ in terms of the weights?

2) I was not expecting to find a $-1$-curve supporting all the quotient singularities of $S$. Is this a common phenomenon? Should I check for $-1$-curves on every weighted hypersurface ?

3) For a weighted hypersurface $S$, to have amplitude zero (i.e $\omega_S =\mathcal{O}_{S}$) means that the surface $S$ is a K3 one. There are 95 of those classified by Reid. Is there a classification of $K3$ surfaces with positive amplitude?

4) The canonical divisor of the K3 surface $S$ is $(x^2+y^3+z^{11}=0)$..any comment?

I will appreciate any idea or suggestion.

[1] "K3-fibered Calabi-Yau threefolds I, the twist map" Bruce Hunt, Rolf Schimmrigk, http://arxiv.org/abs/math/9904059

  • $\begingroup$ What is your definition of a K3 surface? The usual definition is smooth simply connected with trivial canonical bundle (hence nef), so it is minimal by definition. Do you allow surfaces whose minimal models are of this type? $\endgroup$ Nov 14, 2013 at 20:25
  • $\begingroup$ I edited it. I mean that $S$ is birational is to $K3$ surface $\endgroup$
    – eventually
    Nov 14, 2013 at 21:03
  • $\begingroup$ This should be presumably generalized to $dim 3$ lattice polytopes with unique interior lattice point -- the corresponding hypersurface should be birational to $K3$. $\endgroup$ Nov 14, 2013 at 23:57
  • $\begingroup$ @LevBorisov May you elaborate a bit? Is the lattice polytope inducing an ambient toric variety? Is the unique lattice point inducing a hypersurface? ..why is it birational to a K3 $\endgroup$
    – eventually
    Nov 21, 2013 at 3:42


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