Advertisement

Acta Mathematica Vietnamica

, Volume 44, Issue 1, pp 101–116 | Cite as

Correspondence Scrolls

  • David EisenbudEmail author
  • Alessio Sammartano
Article
  • 27 Downloads

Abstract

This paper initiates the study of a class of schemes that we call correspondence scrolls, which includes the rational normal scrolls and linearly embedded projective bundle of decomposable bundles, as well as degenerate K3 surfaces, Calabi-Yau threefolds, and many other examples.

Keywords

Rational normal scroll Veronese embedding Join variety Multiprojective space Variety of complexes Variety of minimal degree Double structure K3 surface Calabi-Yau scheme Gorenstein ring Gröbner basis 

Mathematics Subject Classification (2010)

Primary 14J40 Secondary 13H10 13C40 13P10 14J26 14J28 14J32 14M05 14M12 14M20 

Notes

Acknowledgements

The first author was partially supported by NSF grant No. 1502190. He would like to thank Frank-Olaf Schreyer, who pointed out in their joint work that the K3 carpets could be regarded as coming from correspondences. The second author was supported by NSF grant No. 1440140 while he was a Postdoctoral Fellow at the Mathematical Sciences Research Institute in Berkeley, CA. He would like to thank Aldo Conca and Matteo Varbaro for some helpful comments.

References

  1. 1.
    Alper, J., Fedorchuk, M., Smyth, D.I.: Finite Hilbert stability of (bi) canonical curves. Invent. Math. 191(3), 671–718 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bayer, D., Eisenbud, D.: Ribbons and their canonical embeddings. Trans. Am. Math. Soc. 347(3), 719–756 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blum, S.: Subalgebras of bigraded Koszul algebras. J. Algebra 242(2), 795–809 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bruns, W., Herzog, J.: On the computation of a-invariants. Manuscripta Math. 77(2–3), 201–213 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Conca, A., De Negri, E., Gorla, E: Cartwright-sturmfels ideals associated to graphs and linear spaces. To appear in. J. Combinatorial Algebra, arXiv 1705, 00575 (2017)zbMATHGoogle Scholar
  6. 6.
    De Concini, C., Strickland, E.: On the variety of complexes. Adv. Math. 41, 56–77 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deopurkar, A.: The canonical syzygy conjecture for ribbons. Math. Z. 288(3–4), 1157–1164 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Deopurkar, A., Fedorchuk, M., Swinarski, D.: Gröbner techniques and ribbons. Albanian. J. Math. 8(1), 55–70 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Diaconis, P., Eisenbud, D., Sturmfels, B.: Lattice Walks and Primary Decomposition. Mathematical Essays in Honor of Gian-Carlo Rota (Cambridge, MA, 1996). Progr. Math. 161, 173–193 (1998). Birkhäuser Boston, Boston, MAGoogle Scholar
  10. 10.
    Eisenbud, D.: On the Resiliency of Determinantal Ideals. Commutative Algebra and Combinatorics (Kyoto, 1985), North-Holland, Amsterdam. Adv. Stud Pure Math 11, 29–38 (1987)CrossRefGoogle Scholar
  11. 11.
    Eisenbud, D., Green, M., Hulek, K., Popescu, S.: Small schemes and varieties of minimal degree. Am. J. Math. 128(6), 1363–1389 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Eisenbud, D., Harris, J.: On varieties of minimal degree (a centennial account). Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985). Proc. Sympos. Pure Math 46(1), 3–13 (1987). Am. Math. Soc., Providence RICrossRefGoogle Scholar
  13. 13.
    Eisenbud, D., Schreyer, F.O.: Equations and syzygies of K3 carpets and unions of scrolls. arXiv:1804.08011 (2018)
  14. 14.
    Gallego, F., Purnaprajna, B.: Degenerations of k3 surfaces in projective space. Trans. Am. Math. Soc. 349(6), 2477–2492 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gallego, F., Purnaprajna, B.: On the canonical rings of covers of surfaces of minimal degree. Trans. Am. Math. Soc. 355(7), 2715–2732 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Goto, S., Watanabe, K.: On graded rings. II (z n-graded rings). Tokyo J. Math 1(2), 237–261 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Harris, J.: Curves in Projective Space. With the Collaboration of David Eisenbud Séminaire De Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 85. Presses de l’Université de Montréal, Montreal (1982)Google Scholar
  18. 18.
    Hochster, M., Eagon, J.A.: Cohen-macaulay rings, invariant theory, and the generic perfection of determinantal loci. Am. J. Math. 93, 1020–1058 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hosten, S., Sullivant, S.: Ideals of adjacent minors. J. Algebra 277(2), 615–642 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Manolache, N.: Double rational normal curves with linear syzygies. Manuscripta Math. 104(4), 503–517 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nagel, U., Notari, R., Spreafico, M.L.: Curves of degree two and ropes on a line: their ideals and even liaison classes. J. Algebra 265(2), 772–793 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schreyer, F.O.: Syzygies of canonical curves and special linear series. Math. Ann. 275(1), 105–137 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shibuta, T.: Gröbner bases of contraction ideals. J. Algebraic Combin. 36(1), 1–19 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sturmfels, B.: Gröbner bases and convex Polytopes University Lecture Series, vol. 8. American Mathematical Society, Providence (1996)Google Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Mathematical Sciences Research InstituteBerkeleyUSA
  2. 2.Department of MathematicsUniversity of Notre DameNotre DameUSA

Personalised recommendations