The discriminant and the determinant of a hypersurface of even dimension
Let \(X\) be a proper smooth variety of even dimension \(n\) over a field \(k\) and the characteristic of \(k\) is not \(2\). Further, let \(X\) be a smooth hypersurface of degree \(d\) in a projective space of dimension \(n+1\) defined by a homogeneous polynomial \(f\). Let \(\mathrm{disc}_{d}(f)\) be the divided discriminant of \(f\) (Definition 2.2). The author's main result states that the quadratic character of the absolute Galois group defined by the determinant of the \(\ell\)-adic cohomology of middle dimension is defined by the square root of \(\varepsilon (n,d)\mathrm{disc}_{d}(f)\), where \(\varepsilon (n,d)=(-1)^{\frac{ d-1}{2}}\) if \(d\) is odd and \(\varepsilon (n,d)=(-1)^{\frac{d}{2}\frac{n+2}{2}}\) if \(d\) is even.
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