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arXiv 2004.08303math.AGmath.CV
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阿贝尔簇中一般超曲面的典范映射

Canonical Maps of general Hypersurfaces in Abelian Varieties

Fabrizio Catanese, Luca Cesarano

备注：
11 pages, dedicated to Olivier Debarre on the occasion of his 60th birthday. Final verion, to appear in a special volume of ERA dedicatee to birational geometry
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摘要：

本文的主要定理是：对于阿贝尔簇$A$中的（示例）超曲面$X$的一般对$(A,X)$，如果$X$给出的极化不是主极化，则$X$的正则映射$Φ_X$在其象上是双有理的（即，它的Pfalan $d$不等于$1$）。我们还表明，设置$g = dim (A)$，并让$d$是Pfumeran的极化$X$，那么如果$X$是光滑的和$$Φ_X : X \rightarrow \mathbb{P}^{N:=g+d-2}$$是一个嵌入，那么我们一定有不等式$ d \geq g + 1$，相当于$N : = g+d-2 \geq 2 \ dim(X) + 1.$我们还制定了以下有趣的猜想，动机的工作的第二作者：如果$ d \geq g + 1,$然后，对于一般对$(A,X)$，$Φ_X$是一个嵌入。

英文摘要：

The main theorem of this paper is that, for a general pair $(A,X)$ of an (ample) Hypersurface $X$ in an Abelian Variety $A$, the canonical map $Φ_X$ of $X$ is birational onto its image if the polarization given by $X$ is not principal (i.e., its Pfaffian $d$ is not equal to $1$). We also show that, setting $g = dim (A)$, and letting $d$ be the Pfaffian of the polarization given by $X$, then if $X$ is smooth and $$Φ_X : X \rightarrow \mathbb{P}^{N:=g+d-2}$$ is an embedding, then necessarily we have the inequality $ d \geq g + 1$, equivalent to $N : = g+d-2 \geq 2 \ dim(X) + 1.$ We also formulate the following interesting conjecture, motivated by work of the second author: if $ d \geq g + 1,$ then, for a general pair $(A,X)$, $Φ_X$ is an embedding.

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