December  2021, 29(6): 4315-4325. doi: 10.3934/era.2021087

Canonical maps of general hypersurfaces in Abelian varieties

Lehrstuhl Mathematik VIII, Mathematisches Institut der Universität Bayreuth, NW II, Universitätsstr. 30, 95447 Bayreuth, Germany

* Corresponding author

Dedicated to Olivier Debarre on the occasion of his 60-th +$\epsilon$ birthday

Received  December 2020 Revised  September 2021 Published  December 2021 Early access  October 2021

Fund Project: The present work took place in the framework of the ERC Advanced grant n. 340258, 'TADMICAMT'

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. Electronic Research Archive, 2021, 29 (6) : 4315-4325. doi: 10.3934/era.2021087
References:
[1]

W. Barth, Abelian surfaces with (1, 2)-polarization, Adv. Stud. Pure Math., 10 (1987), 41-84.  doi: 10.2969/aspm/01010041.  Google Scholar

[2]

F. Catanese, On Severi's proof of the double point formula, Comm. Algebra, 7 (1979), 763-773.  doi: 10.1080/00927877908822373.  Google Scholar

[3]

F. Catanese, Pluricanonical Gorenstein curves, Progr. Math., Birkhäuser Boston, Boston, MA, 24 (1982), 51-95.   Google Scholar

[4]

F. Catanese and F.-O. Schreyer, Canonical projections of irregular algebraic surfaces, Algebraic Geometry, (2002), 79–116.  Google Scholar

[5]

F. Catanese and K. Oguiso, The double point formula with isolated singularities and canonical embeddings, J. Lond. Math. Soc., 102 (2020), 1337-1356.  doi: 10.1112/jlms.12371.  Google Scholar

[6]

L. Cesarano, Canonical Surfaces and Hypersurfaces in Abelian Varieties, arXiv: 1808.05302. Google Scholar

[7]

G. CodogniS. Grushevsky and E. Sernesi, The degree of the Gauss map of the theta divisor, Algebra Number Theory, 11 (2017), 983-1001.  doi: 10.2140/ant.2017.11.983.  Google Scholar

[8]

O. Debarre, Le lieu des variétés abéliennes dont le diviseur theta est singulier a deux composantes, [The locus of abelian varieties whose theta divisor is singular has two components], Ann. Sci. École Norm. Sup., 4 (1992), 687-707.  doi: 10.24033/asens.1664.  Google Scholar

[9]

O. DebarreK. Hulek and J. Spandaw, Very ample linear systems on abelian varieties, Math. Ann., 300 (1994), 181-202.  doi: 10.1007/BF01450483.  Google Scholar

[10]

W. Fulton and D. Laksov, Residual intersections and the double point formula, Real and complex singularities, Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976, Sijthoff and Noordhoff, Alphen aan den Rijn, (1977), 171–177.  Google Scholar

[11]

H. Grauert and R. Remmert, Komplexe Räume [Complex Spaces], Math. Ann., 136 (1958), 245-318.  doi: 10.1007/BF01362011.  Google Scholar

[12]

M. L. Green, Quadrics of rank four in the ideal of a canonical curve, Invent. Math., 75 (1984), 85-104.  doi: 10.1007/BF01403092.  Google Scholar

[13]

S. Lefschetz, On certain numerical invariants of algebraic varieties with application to abelian varieties, Trans. Amer. Math. Soc., 22 (1921), 327-406.  doi: 10.1090/S0002-9947-1921-1501178-3.  Google Scholar

[14]

D. S. Nagaraj and S. Ramanan, Polarisations of type $(1, 2\dots, 2)$ on abelian varieties, Duke Math. J., 80 (1995), 157-194.  doi: 10.1215/S0012-7094-95-08007-7.  Google Scholar

[15]

A. Ohbuchi, Some remarks on ample line bundles on abelian varieties, Manuscripta Math., 57 (1987), 225-238.  doi: 10.1007/BF02218082.  Google Scholar

[16]

M. Penegini and F. Polizzi, A new family of surfaces with $p_g = q = 2$, and $K^2 = 6$ whose Albanese map has degree 4, J. Lond. Math. Soc., 90 (2014), 741-762.  doi: 10.1112/jlms/jdu048.  Google Scholar

[17]

G. P. Pirola, Curves on generic Kummer varieties, Duke Math. J., 59 (1989), 701-708.  doi: 10.1215/S0012-7094-89-05931-0.  Google Scholar

[18]

Z. Ran, The structure of Gauss-like maps, Compositio Math., 52 (1984), 171-177.   Google Scholar

[19]

F. Severi, Sulle intersezioni delle varietá algebriche e sopra i loro caratteri e singolaritá proiettive, Mem. Accad. Scienze di Torino, S. II, also in Memorie Scelte I, Zuffi (1950) (Bologna), 52 (1902), 61–118. Google Scholar

[20]

A. van de Ven, On the embeddings of abelian varieties in projective space, Ann. Mat. Pura Appl., 103 (1975), 127-129.  doi: 10.1007/BF02414149.  Google Scholar

show all references

References:
[1]

W. Barth, Abelian surfaces with (1, 2)-polarization, Adv. Stud. Pure Math., 10 (1987), 41-84.  doi: 10.2969/aspm/01010041.  Google Scholar

[2]

F. Catanese, On Severi's proof of the double point formula, Comm. Algebra, 7 (1979), 763-773.  doi: 10.1080/00927877908822373.  Google Scholar

[3]

F. Catanese, Pluricanonical Gorenstein curves, Progr. Math., Birkhäuser Boston, Boston, MA, 24 (1982), 51-95.   Google Scholar

[4]

F. Catanese and F.-O. Schreyer, Canonical projections of irregular algebraic surfaces, Algebraic Geometry, (2002), 79–116.  Google Scholar

[5]

F. Catanese and K. Oguiso, The double point formula with isolated singularities and canonical embeddings, J. Lond. Math. Soc., 102 (2020), 1337-1356.  doi: 10.1112/jlms.12371.  Google Scholar

[6]

L. Cesarano, Canonical Surfaces and Hypersurfaces in Abelian Varieties, arXiv: 1808.05302. Google Scholar

[7]

G. CodogniS. Grushevsky and E. Sernesi, The degree of the Gauss map of the theta divisor, Algebra Number Theory, 11 (2017), 983-1001.  doi: 10.2140/ant.2017.11.983.  Google Scholar

[8]

O. Debarre, Le lieu des variétés abéliennes dont le diviseur theta est singulier a deux composantes, [The locus of abelian varieties whose theta divisor is singular has two components], Ann. Sci. École Norm. Sup., 4 (1992), 687-707.  doi: 10.24033/asens.1664.  Google Scholar

[9]

O. DebarreK. Hulek and J. Spandaw, Very ample linear systems on abelian varieties, Math. Ann., 300 (1994), 181-202.  doi: 10.1007/BF01450483.  Google Scholar

[10]

W. Fulton and D. Laksov, Residual intersections and the double point formula, Real and complex singularities, Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976, Sijthoff and Noordhoff, Alphen aan den Rijn, (1977), 171–177.  Google Scholar

[11]

H. Grauert and R. Remmert, Komplexe Räume [Complex Spaces], Math. Ann., 136 (1958), 245-318.  doi: 10.1007/BF01362011.  Google Scholar

[12]

M. L. Green, Quadrics of rank four in the ideal of a canonical curve, Invent. Math., 75 (1984), 85-104.  doi: 10.1007/BF01403092.  Google Scholar

[13]

S. Lefschetz, On certain numerical invariants of algebraic varieties with application to abelian varieties, Trans. Amer. Math. Soc., 22 (1921), 327-406.  doi: 10.1090/S0002-9947-1921-1501178-3.  Google Scholar

[14]

D. S. Nagaraj and S. Ramanan, Polarisations of type $(1, 2\dots, 2)$ on abelian varieties, Duke Math. J., 80 (1995), 157-194.  doi: 10.1215/S0012-7094-95-08007-7.  Google Scholar

[15]

A. Ohbuchi, Some remarks on ample line bundles on abelian varieties, Manuscripta Math., 57 (1987), 225-238.  doi: 10.1007/BF02218082.  Google Scholar

[16]

M. Penegini and F. Polizzi, A new family of surfaces with $p_g = q = 2$, and $K^2 = 6$ whose Albanese map has degree 4, J. Lond. Math. Soc., 90 (2014), 741-762.  doi: 10.1112/jlms/jdu048.  Google Scholar

[17]

G. P. Pirola, Curves on generic Kummer varieties, Duke Math. J., 59 (1989), 701-708.  doi: 10.1215/S0012-7094-89-05931-0.  Google Scholar

[18]

Z. Ran, The structure of Gauss-like maps, Compositio Math., 52 (1984), 171-177.   Google Scholar

[19]

F. Severi, Sulle intersezioni delle varietá algebriche e sopra i loro caratteri e singolaritá proiettive, Mem. Accad. Scienze di Torino, S. II, also in Memorie Scelte I, Zuffi (1950) (Bologna), 52 (1902), 61–118. Google Scholar

[20]

A. van de Ven, On the embeddings of abelian varieties in projective space, Ann. Mat. Pura Appl., 103 (1975), 127-129.  doi: 10.1007/BF02414149.  Google Scholar

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