2019/03/12 by Helge Øystein Maakestad, Maakestad, Helge Øystein
Mathematics · #14E05 #14E08 #14E30 #14N05 #14N15 #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1903.04871
openalex publication_date 2019/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The aim of this note is to use elementary methods to give a large class of examples of projective varieties Y ⊆ ℙdk over a field k with the property that Y is not isomorphic to a hypersurface H⊆ ℙNk in projective space ℙNk with N:=dim(Y)+1. We apply this construction to the study of the arithmetic genus pa(Y) of Y and the problem of determining if pa(Y) is a birational invariant of Y in general. We give an infinite number of examples of pairs of projective varieties (Y, Y') in any dimension dim(Y)=dim(Y')≥ 4 where Y is birational to Y', but where pa(Y)≠ pa(Y'). The arithmetic genus is by Hodge theory known to be a birational invariant for smooth projective varieties over an algebraically closed field of characteristic zero. In each dimension d≥ 4 we give positive dimensional families of pairs of projective varieties (Y,Y') that are birational but where the arithmetic genus differ. We prove a similar result on the Euler characteristic.