Linear families of smooth hypersurfaces over finitely generated fields

2022/07/02 by Asgarli, Shamil, Ghioca, Dragos, Reichstein, Zinovy
#14G15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 14N05 #Secondary 14J70

paper · doi:10.48550/arxiv.2207.00918

Abstract

Let K be a finitely generated field. We construct an n-dimensional linear system L of hypersurfaces of degree d in ℙⁿ defined over K such that each member of L defined over K is smooth, under the hypothesis that the characteristic p does not divide gcd(d, n+1) (in particular, there is no restriction when K has characteristic 0). Moreover, we exhibit a counterexample when p divides gcd(d, n+1).

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