### Electronic Research Archive

2021, Issue 6: 4315-4325. doi: 10.3934/era.2021087
Special Issues

# Canonical maps of general hypersurfaces in Abelian varieties

• Received: 01 December 2020 Revised: 01 September 2021 Published: 26 October 2021
• 14E05, 14E25, 14M99, 14K25, 14K99, 14H40, 32J25, 32Q55, 32H04

• 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 $\Phi_X$ 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 easily show that, setting $g = dim (A)$, and letting $d$ be the Pfaffian of the polarization given by $X$, then if $X$ is smooth and

$\Phi_X : X {\rightarrow } {\mathbb{P}}^{N: = g+d-2}$

is an embedding, then necessarily we have the inequality $d \geq g + 1$, equivalent to $N : = g+d-2 \geq 2 \ dim(X) + 1.$

Hence we formulate the following interesting conjecture, motivated by work of the second author: if $d \geq g + 1,$ then, for a general pair $(A,X)$, $\Phi_X$ is an embedding.

Citation: Fabrizio Catanese, Luca Cesarano. Canonical maps of general hypersurfaces in Abelian varieties[J]. Electronic Research Archive, 2021, 29(6): 4315-4325. doi: 10.3934/era.2021087

### Related Papers:

• 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 $\Phi_X$ 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 easily show that, setting $g = dim (A)$, and letting $d$ be the Pfaffian of the polarization given by $X$, then if $X$ is smooth and

$\Phi_X : X {\rightarrow } {\mathbb{P}}^{N: = g+d-2}$

is an embedding, then necessarily we have the inequality $d \geq g + 1$, equivalent to $N : = g+d-2 \geq 2 \ dim(X) + 1.$

Hence we formulate the following interesting conjecture, motivated by work of the second author: if $d \geq g + 1,$ then, for a general pair $(A,X)$, $\Phi_X$ is an embedding.

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