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Orbits of Curves on Certain K3 Surfaces

Authors:

Arthur Baragar

Affiliation:

(1) Department of Mathematical Science, University of Nevada, Box 454020, 4505 Maryland Parkway, Las Vegas, NV, 89154-4020, U.S.A.

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< agr

<.6538.

Keywords:

fractal dimension K3 surface Picard group

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