Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On the birational invariance of the arithmetic genus and Euler characteristic (1903.04871v8)

Published 12 Mar 2019 in math.AG

Abstract: The aim of this note is to use elementary methods to give a large class of examples of projective varieties $ Y \subseteq \mathbb{P}d_k$ over a field $k$ with the property that $Y$ is not isomorphic to a hypersurface $H\subseteq \mathbb{P}N_k$ in projective space $\mathbb{P}N_k$ with $N:=dim(Y)+1$. We apply this construction to the study of the arithmetic genus $p_a(Y)$ of $Y$ and the problem of determining if $p_a(Y)$ 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 $dim(Y)=dim(Y')\geq 4$ where $Y$ is birational to $Y'$, but where $p_a(Y)\neq p_a(Y')$. 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 $d\geq 4$ 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.

Summary

We haven't generated a summary for this paper yet.