arXiv 1903.048712019-03math.AG

算术亏格的双有理不变性与欧拉示性数

On the birational invariance of the arithmetic genus and Euler characteristic

Helge Øystein Maakestad

备注：: Corollary 2.8: Minor correction. 25.04:2019: A reference and a result on the Euler characteristic added. 11.05.2019: Minor corrections. 29.05.2019: Example 2.11 added. 03.06.2019: Theorem 2.12 added

摘要：

本文的目的是用初等方法给出域$k$上的射影簇$ Y \subseteq \mathbb{P}^d_k$的一大类例子，这些例子具有这样的性质：$Y$不同构于具有$N:=dim(Y)+1$的射影空间$\mathbb{P}^N_k$中的超曲面$H\subseteq \mathbb{P}^N_k$.我们应用这个构造研究$Y$的算术亏格$p_a(Y)$和确定$p_a(Y)$是否是一般$Y$的双有理不变量的问题。我们给出了无限多的例子对投影品种$(Y, Y')$在任何维度$dim(Y)=dim(Y')\geq 4$ $Y$是双有理$Y'$，但其中$p_a(Y)\neq p_a(Y')$。算术亏格是由霍奇理论已知是一个双有理不变量的光滑射影簇上的代数闭域的特征零。在每个维度$d\geq 4$，我们给正维家庭的对投影品种$(Y,Y')$是双有理的，但在算术亏格不同。我们证明了关于欧拉示性数的一个类似结果。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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