Orbits of Curves on Certain K3 Surfaces

TLDR: In this paper, the authors studied the family of algebraic K3 surfaces generated by the smooth intersection of a (1, 1) form and a (2, 2) form in the Picard number 3.

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 \(\mathbb{P}^{2} { \times }\mathbb{P}^{2} \) defined over \(\mathbb{C}\) and with Picard number 3. We describe the group of automorphisms \(\mathcal{A} = Aut (V / {\mathbb{C}})\) on V. For an ample divisor D and an arbitrary curve C0 on V, we investigate the asymptotic behavior of the quantity \(N_{\mathcal{A}{\text{(}}C_0 {\text{)}}} (t) = \# \{ C \in \mathcal{A}{\text{(}}C_0 {\text{)}}\;{\text{:}}\;C \cdot D < t\} \). We show that the limit $$\mathop {\lim }\limits_{t \to \infty } \frac{{\log N_{\mathcal{A}(C)} (t)}}{{\log t}} = \alpha $$ exists, does not depend on the choice of curve C or ample divisor D, and that .6515<α<.6538.