ScalingStacks

Example 3.11. For generic quartic K3 surfaces, the following simple situation models a small neighbourhood of the 24 points on $\text{K3}\cap\{Z_{i}=Z_{j}=0\}$. Locally the dominant monomials are $(z_{1}z_{2})^{-1},(z_{1}z_{2})^{-1}z_{0},1$, where $z_{1},z_{2}$ are $\mathbb{C}$-coodinates which vanish on toric boundaries, and $z_{0}$ is a $\mathbb{C}^{*}$-coordinate; together $z_{0},z_{1},z_{2}$ are local coordinates on $\mathbb{P}^{3}$. The local model hypersurface is

$$\{-(z_{1}z_{2})^{-1}+(z_{1}z_{2})^{-1}z_{0}=1\}=\{z_{0}=1+z_{1}z_{2}\},$$

so $z_{1},z_{2}$ can be used as local coordinates on the hypersurface. The holomorphic volume form $\Omega$ on the hypersurface is (up to a normalising factor)

$$\Omega=\frac{d\log z_{0}\wedge d\log z_{1}\wedge d\log z_{2}}{d(-(z_{1}z_{2})^{-1}+(z_{1}z_{2})^{-1}z_{0}-1)}=z_{0}^{-1}dz_{1}\wedge dz_{2}.$$

This is the typical boundary type behaviour. A significant part of the boundary type region overlaps with the toric region. In this example, when $|z_{1}|$ is not too small, we can view $z_{1}$ as a $\mathbb{C}^{*}$-coordinates, so $\{z_{1},z_{0}\}$ provides a toric type chart, as we can express $z_{2}=z_{1}^{-1}(z_{0}-1)$. In this chart

$$\Omega=z_{0}^{-1}dz_{1}\wedge dz_{2}=d\log z_{1}\wedge d\log z_{0},$$

which agrees with the standard holomorphic volume form in toric type charts. The same behaviour happens when $|z_{2}|$ is not too small. The problem mentioned in Remark 3.10 is due to the fact that this local model is only a valid approximate description of the K3 for $z_{0},z_{1},z_{2}$ satisfying some inequality constraints. The prescription of charts of boundary type means that we are simultaneously using the charts $\{|z_{1}|\lesssim\nu,|z_{2}|\lesssim\nu^{-1}\}$ for many choices of parameters $\nu$. Notice the scaling symmetry

$$z_{1}\mapsto\nu z_{1},\quad z_{2}\mapsto\nu^{-1}z_{2}$$

means that there is no obviously preferred chart of boundary type. More concrete examples can be found in [30, section 1.1.6].