The discriminant of a hypersurface in weighted projective space

Abstract

In this paper, we study the discriminant of a hypersurface in a weighted projective space with isolated singularities. We define the discriminant of a hypersurface in [Formula: see text] with [Formula: see text], and show that it satisfies a smoothness criterion over any commutative ring. A smooth hypersurface of degree [Formula: see text] in the weighted projective three-fold [Formula: see text] is a del Pezzo surface of degree [Formula: see text]. We show that the determinant of the Galois action on the [Formula: see text]-lattice arising from its exceptional curves can be computed via the square roots of the discriminant.