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)
"Algebraic Number Theory and Related Topics 2013". December 9～13, 2013. edited by Tadashi Ochiai, Takeshi Tsuji and Iwao Kimura. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.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
