# On the birational invariance of the arithmetic genus and Euler characteristic

## TL;DR

This paper constructs examples of projective varieties that are birational but have different arithmetic genus and Euler characteristic, challenging the invariance of these invariants beyond smooth cases.

## Contribution

It provides explicit examples of birational varieties with differing arithmetic genus and Euler characteristic, extending understanding of their invariance properties.

## Key findings

- Examples of birational varieties with different arithmetic genus in all dimensions ≥ 4
- Counterexamples to birational invariance of Euler characteristic
- Construction of large classes of non-hypersurface varieties

## 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.