$A_{N-1}$ Singularities in a K3 Surface? I have some understanding of the typical $A_{N-1}$ singularities arising as the orbifold $\mathbb{C}^{2}/\mathbb{Z}_{N}$.  Of course, you can blow up the origin to get a bunch of rigid, rational curves linked together just as in the Dynkin diagram.  I've seen in a few places people discussing "$A_{N-1}$ singularities inside a K3 surface."  
I was hoping someone could explain to me how such a thing is constructed.  I know $\mathbb{C}^{2}$ is a Calabi-Yau just like a K3 surface, so do you just naïvely take an orbifold quotient of a K3 surface by $\mathbb{Z}_{N}$, and then blow up to get a smooth Calabi-Yau with a bunch of exceptional $\mathbb{P}^{1}$'s?  I'm unsure how to define this group action though; it is clean in the case of the plane.  
Also, can this be done with any K3 surface? I know the rational curves in a K3 are rigid (i.e. they move in a zero dimensional family) and in particular, I think not every K3 even contains rational curves. So I guess if you take one of these quotients of a K3 without rational curves, you blow up to arrive at a smooth K3 which indeed has rational curves?  
(BTW, this came to me in the context of physics: apparently, in Type IIA string theory, if you compactify on such a K3 as above, after blowing up the singularities you get a $\mathcal{N}=4\,U(N)$ gauge theory in six dimensions.)
 A: One good example is the Kummer K3 surface. Start with a complex two-torus $\mathbb C^2 / \Lambda$, where $\Lambda$ is a lattice of rank four. Then quotient by the $\mathbb Z_2$ automorphism:
$$ (x, y) \mapsto (-x, -y),$$
where $x, y \in \mathbb C$ are the natural complex coordinates on the two-torus.
This automorphism has 16 isolated fixed points. After performing the quotient, you are left with 16 $A_1$ singularities at those fixed points.
If you blow up those singularities, which resolves them into rational curves with self-intersection $-2$, you end up with a smooth K3 surface. Indeed, it is simple to check that the canonical divisor remains trivial after the resolution.
Of course, such a K3 surface is very special. Notably, its Picard rank is at least 16. The generic (non-algebraic) K3 has Picard rank zero. So you shouldn't expect a "neighbouring point in the K3 moduli space" to also be of Kummer type.
If you start with a two-torus of a special shape, e.g. a product of two elliptic curves with complex structure parameter $\tau = i$ or $\exp(2\pi i /3)$, it's actually possible to quotient by discrete groups larger than $\mathbb Z_2$ to get singular K3s. You can end up with orbifold singularities of $A_2$, $A_3$, $A_5$, $D_4$, $D_5$ and $E_6$ type. Katrin Wendland has classified all the possibilities in this very nice physics paper: https://arxiv.org/abs/hep-th/0010281.
