The determinant of a double covering of the projective space and the discriminant of the branch locus : announcement (Algebraic Number Theory and Related Topics 2013)

Abstract

The determinant of the Galois action on the ell-adic cohomology of the middle degree of a proper smooth variety of even dimension defines a quadratic character of the absolute Galois group of the base field of the variety. In this announcement, we state that for a double covering of the projective space of even dimension, the character is computed via the square root of the discriminant of the defining polynomial of the covering. As a corollary, we deduce that the parity of a Galois permutation of the exceptional divisors on a del Pezzo surface can be computed by the discriminant.