Let p be a prime integer, and q a power of p. The Ballico-Hefez curve is a non-reflexive nodal rational plane curve of degree q + 1 in characteristic p. We investigate its automorphism group and defining equation. We also prove that the surface obtained as the cyclic cover of the projective plane branched along the Ballico-Hefez curve is unirational, and hence is supersingular. As an application, we obtain a new projective model of the supersingular K3 surface with Artin invariant 1 in characteristic 3 and 5.

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We work over an algebraically closed field k of positive characteristic p. Let q be a power of p. Let A be an (n + 1) × (n + 1) matrix with coefficients a ij in k, and let X A be a hypersurface of degree q + 1 in the projective space P n defined by a ij x i x q j = 0. It is well-known that if the rank of A is n + 1, the hypersurface X A is projectively isomorphic to the Fermat hypersuface of degree q + 1. We investigate the hypersurfaces X A when the rank of A is n, and determine their projective isomorphism classes.