The discriminant and the determinant of a hypersurface of even dimension
type: Item
instance of: scholarly article
title: The discriminant and the determinant of a hypersurface of even dimension
zbMATH Open document ID: 1285.14046
author: Takeshi Saito
published in: Mathematical Research Letters
publication date: 2013-05-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1110.1717
review text: 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.
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 6165858
zbMATH Keywords: variety
zbMATH Keywords: hypersurface
zbMATH Keywords: cohomology
zbMATH Keywords: absolute Galois group
zbMATH Keywords: quadratic character
zbMATH Keywords: divided discriminant
reviewed by: Igor G. Nikolaev
MaRDI profile type: Publication
arXiv ID: 1110.1717
DOI: 10.4310/MRL.2012.V19.N4.A10
description: scientific article