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
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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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
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