A Simple Proof That Rational Curves on K3 Are Nodal

The Picard group of S is generated by two effective divisors C and F with C = −2, F 2 = 0 and C · F = 1. It can be realized as an elliptic fiberation over P with a unique section C, fibers F and λ = 2. It is the same special K3 surface used by Bryan and Leung in their counting of curves on K3 surfaces [B-L]. It is actually the attempt to understand their method that leads us to our proof. We will call a K3 surface with Picard lattice (1.1) a BL K3 surface. A BL K3 surface S lies on the boundary of the moduli space of K3 surfaces of genus g with C+gF as the corresponding primitive divisor. Every curve in the linear series |OS(C + gF )| is “totally reducible”, i.e., it consists of the −2 curve C and g elliptic “tails” attached to C. A curve D ∈ |OS(C + gF )| is the image of a stable rational map only if D = C ∪m1F1∪m2F2∪ ...∪m24F24, where F1, F2, ..., F24 are 24 rational nodal curves in the pencil |F | and ∑24 i=1mi = g; D is obviously nodal if mi ≤ 1 for all i. The main problem is, of course, mi might be greater than 1, i.e., D might be nonreduced, in which case we need to show that when S deforms to a general K3 surface S ′ of genus g and D