Rational curves on the supersingular K3 surface with Artin invariant 1 in characteristic 3

Abstract

We show the existence of 112 non-singular rational curves on the supersingular K3 surface X with Artin invariant 1 in characteristic 3 by several ways. These non-singular rational curves have the minimum degree with respect to a very ample divisor on X. Using these rational curves, we have a ( 16 ) 10 -configuration and a ( 280 4 , 112 10 ) -configuration on the K3 surface. Moreover we study the Picard lattice by using the theory of the Leech lattice. The 112 non-singular rational curves correspond to 112 Leech roots.