Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the birational invariance of the arithmetic genus and Euler
  characteristic   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.

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The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
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The categories of this paper are: math-ph