K3 surface criteria

Question

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?

Answer 1

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.

Answer 2

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