K3 surface criteria Suppose I have an affine equation $f(x, y) = 0$ which after homogenizing becomes $f(X, Y, Z) = 0$ in $\mathbb{P}^{3}$. Are there ways to check that $f$ represents a K3 surface?
 A: I am convinced that $f(X,Y,Z)$ represents a K3 surface iff it is a quartic (fourth-degree) polynomial, see page 16 of this review of K3 surfaces:

http://arxiv.org/abs/hep-th/9611137

A: There sure are. Let's denote by $X$ the surface defined by $f$, and let's suppose that $X$ is smooth. For any compact complex surface, being K3 is equivalent to being simply connected and having trivial canonical bundle.
A hypersurface in $\mathbb P^n$ is connected and simply connected by the Lefschetz theorem, so we only need to find a condition ensuring that the canonical bundle is trivial.
This condition is given by the adjunction formula, which says that if the polynomial $f$ is of degree $d$, then
$$ K_X = ( K_{\mathbb P^3} \otimes \mathcal O(d) )_{|X} = \mathcal O_X(d-4). $$
This bundle is trivial if and only if $d = 4$, or in other words, if $f$ is a quartic.
A fun exercise involving the adjuction formula is to see that there are very few K3 surfaces given as complete intersections in $\mathbb P^n$. In fact, they only exist in dimension 4, 5 and 6 if I remember correctly.
