Orbits of Curves on Certain K3 Surfaces

Abstract

In this paper, we study the family of algebraic K3 surfaces generated by the smooth intersection of a (1, 1) form and a (2, 2) form in ${\open P}^2\times{\open P}^2$ defined over ${\open C}$ and with Picard number 3. We describe the group of automorphisms ${\cal A}={\rm Aut}(V/{\open C})$ on V. For an ample divisor D and an arbitrary curve C₀ on V, we investigate the asymptotic behavior of the quantity $N_{{\cal A}(C_0)}(t)=\#\{C \in {\cal A}(C_0): C\cdot D<t\}$. We show that the limit $\lim_{t\to \infty } {\log N_{{\cal A}(C\,)}(t)\over \log t}=\alpha\fleqno{}$ exists, does not depend on the choice of curve C or ample divisor D, and that .6515<α<.6538.