The determinant and the discriminant of a complete intersection of even dimension

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. In this article, we show that for a complete intersection of even dimension in a projective space, the character is computed via the square root of the discriminant of the defining polynomials of the variety.

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Received 7 January 2016

Accepted 28 July 2016

Published 4 June 2018