1) a K3 surface is a complex surface which is smooth, irreducible, with p_{g} = p_{a} = 1.

a) Define p_{g} and p_{a}.

b) Prove that the surface in ℙ^{3} given by

X^{4} + Y^{4} + Z^{4} + W^{4} = 0

is a K3 surface.

c) Show that the product of two elliptic curves is a surface with p_{g} = 1, but not p_{a} = 1.

Answers:

1) p_{g} = dimH^{0}(X, ω_{X}) = dimH^{1}(X, O_{X}). (II 8.18, page 180) If dim X = r, then p_{a}(X) = (−1)^{r}(χ(O_{X} − 1) (III, ex. 5.3, page 230) Where χ is defined (page 230, III ex 5.1), by

χ(F) = ∑(−1)^{i}dimH^{i}(X, F).

2) See III, Ex 5.5, page 231. To show that p_{g} = 0, use the adjunction formular (unfortunately this is prob-

ably not covered - but we could just state it.) Use the fact that ω_{ℙ}^{3} = O(−4) Alternateively, use II Ex 8.4, page 188. This is also in II.8 Example 8.20.3, page 183.