Kyoto Journal of Mathematics

Critical k-very ampleness for abelian surfaces

Wafa Alagal and Antony Maciocia

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Abstract

Let (S,L) be a polarized abelian surface of Picard rank 1, and let ϕ be the function which takes each ample line bundle L' to the least integer k such that L' is k-very ample but not (k+1)-very ample. We use Bridgeland’s stability conditions and Fourier–Mukai techniques to give a closed formula for ϕ(Ln) as a function of n, showing that it is linear in n for n>1. As a by-product, we calculate the walls in the Bridgeland stability space for certain Chern characters.

Article information

Source
Kyoto J. Math., Volume 56, Number 1 (2016), 33-47.

Dates
Received: 11 March 2014
Revised: 22 May 2014
Accepted: 3 December 2014
First available in Project Euclid: 15 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1458047877

Digital Object Identifier
doi:10.1215/21562261-3445147

Mathematical Reviews number (MathSciNet)
MR3479317

Zentralblatt MATH identifier
1342.14024

Subjects
Primary: 14C20: Divisors, linear systems, invertible sheaves
Secondary: 14D22: Fine and coarse moduli spaces 14K99: None of the above, but in this section

Keywords
Abelian surface Bridgeland stability moduli spaces very ample

Citation

Alagal, Wafa; Maciocia, Antony. Critical $k$ -very ampleness for abelian surfaces. Kyoto J. Math. 56 (2016), no. 1, 33--47. doi:10.1215/21562261-3445147. https://projecteuclid.org/euclid.kjm/1458047877


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References

  • [AB] D. Arcara and A. Bertram, “Reider’s theorem and Thaddeus pairs revisited” in Grassmannians, Moduli Spaces and Vector Bundles, Clay Math. Proc. 14, Amer. Math. Soc., Providence, 2011, 51–68.
  • [BS] T. Bauer and T. Szemberg, Primitive higher order embeddings of abelian surfaces, Trans. Amer. Math. Soc. 349, no. 4 (1997), 1675–1683.
  • [BM] A. Bayer and E. Macrì, Projectivity and birational geometry of Bridgeland moduli spaces, J. Amer. Math. Soc. 27 (2014), 707–752.
  • [BFS] M. Beltrametti, P. Francia, and A. J. Sommese, On Reider’s method and higher order embeddings, Duke Math. J. 58 (1989), 425–439.
  • [BS1] M. Beltrametti and A. J. Sommese, “Zero cycles and $k$th order embeddings of smooth projective surfaces” with an appendix by L. Göttsche in Problems in the Theory of Surfaces and their Classification (Cortona, 1988), Sympos. Math. 32, Academic Press, London, 1991, 33–48.
  • [BS2] M. Beltrametti and A. J. Sommese, “On $k$-jet ampleness” in Complex Analysis and Geometry, Univ. Ser. Math., Plenum, New York, 1993, 355–376.
  • [Bri] T. Bridgeland, Stability conditions on ${K}3$ surfaces, Duke Math. J. 141 (2008), 241–291.
  • [Huy1] D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry, Oxf. Math. Monogr., Oxford Univ. Press, Oxford, 2006.
  • [Huy2] D. Huybrechts, Derived and abelian equivalence of ${K}3$ surfaces, J. Algebraic Geom. 17 (2008), 375–400.
  • [HL] D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, 2nd ed., Cambridge Math. Libr., Cambridge Univ. Press, Cambridge, 2010.
  • [Mac] A. Maciocia, Computing the walls associated to Bridgeland stability conditions on projective surfaces, Asian J. Math. 18 (2014), 263–279.
  • [MM] A. Maciocia and C. Meachan, Rank $1$ Bridgeland stable moduli spaces on a principally polarized abelian surface, Int. Math. Res. Not. IMRN 2013, no. 9, 2054–2077.
  • [MYY] H. Minamide, S. Yanagida, and K. Yoshioka, Some moduli spaces of Bridgeland’s stability conditions, Int. Math. Res. Not. IMRN 2014, no. 19, 5264–5327.
  • [Muk] S. Mukai, Semi-homogeneous vector bundles on an Abelian variety, Kyoto J. Math. 18 (1978), 239–272.
  • [Mum] D. Mumford, Abelian Varieties, Tata Inst. Fund. Res. Stud. Math. 5, Oxford Univ. Press, London, 1974.
  • [PP1] G. Pareschi and M. Popa, Regularity on abelian varieties, I, J. Amer. Math. Soc. 16 (2003), 285–302.
  • [PP2] G. Pareschi and M. Popa, Regularity on abelian varieties, II: Basic results on linear series and defining equations, J. Algebraic Geom. 13 (2004), 167–193.
  • [T1] H. Terakawa, Higher order embeddings of algebraic surfaces of Kodaira dimension zero, Math. Z. 229 (1998), 417–433.
  • [T2] H. Terakawa, The $k$-very ampleness and $k$-spannedness on polarized abelian surfaces, Math. Nachr. 195 (1998), 237–250.
  • [YY] S. Yanagida and K. Yoshioka, Bridgeland’s stabilities on abelian surfaces, Math. Z. 276 (2014), 571–610.
  • [Y] K. Yoshioka, Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), 817–884.