Abstract
In this article we study the tropicalization of the Hilbert scheme and its suitability as a parameter space for tropical varieties. We prove that the points of the tropicalization of the Hilbert scheme have a tropical variety naturally associated to them. To prove this, we find a bound on the degree of the elements of a tropical basis of an ideal in terms of its Hilbert polynomial. As corollary, we prove that the set of tropical varieties defined over an algebraically closed valued field only depends on the characteristic pair of the field and the image group of the valuation. In conclusion, we examine some simple examples that suggest that the definition of tropical variety should include more structure than what is usually considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atiyah, M.F., Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, (1969)
Bayer, D.: The division algorithm and the Hilbert scheme. PhD thesis, Harvard University. http://www.math.columbia.edu/~bayer/thesis/thesis.html. AAT8222588 (1982)
Bayer, D., Mumford, D.: What can be computed in algebraic geometry? Computational algebraic geometry and commutative algebra (Cortona, 1991), Sympos. Math., XXXIV, pp. 1–48. Cambridge University Press, Cambridge (1993)
Bogart T., Jensen A., Speyer D., Sturmfels B., Thomas R.: Computing tropical varieties. J. Symbolic Comput. 42(1–2), 54–73 (2007)
Brodsky, S., Sturmfels, B.: Tropical quadrics through three points. arXiv:1002.0603v1 (preprint, 2010)
Bruns W., Herzog J.: Cohen-Macaulay rings. Cambridge University Press, Cambridge (1998)
Engler A.J., Prestel A.: Valued fields. In: Springer Monographs in Mathematics. Springer, Berlin (2005)
Eisenbud, D.: Commutative algebra, with a view toward algebraic geometry. In: Graduate Texts in Mathematics. Springer, New York (1995)
Gelfand I.M., Kapranov M.M., Zelevinsky A.V.: Discriminants, resultants, and multidimensional determinants. In: Mathematics: Theory and Applications. Birkhäuser, Boston (1994)
Gotzmann G.: Eine Bedingung für die Flachheit und das Hilbertpolynom eines gradiuerten Ringes. Math. Z. 158, 61–70 (1978)
Hartshorne R.: Connectedness of the Hilbert scheme. Publ. Math. IHÉS 29, 5–48 (1966)
Hept K., Theobald T.: Tropical bases by regular projections. Proc. Am. Math. Soc. 137(7), 2233–2241 (2009)
Sernesi E.: Deformations of algebraic schemes. In: Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (2006)
Maclagan, D., Sturmfels, B.: Introduction to tropical geometry. Draft for a textbook, version October 9 (2009)
Sturmfels B., Trung N.V., Vogel W.: Bounds on degrees of projective schemes. Math. Ann. 302, 417–432 (1995)
Speyer D.: Tropical Geometry. Ph.D. thesis. University of California, Berkeley. http://www-personal.umich.edu/~speyer/thesis.pdf. AAT3187158 (2005)
Speyer D., Sturmfels B.: The tropical Grassmannian. Adv. Geom. 4(3), 389–411 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alessandrini, D., Nesci, M. On the tropicalization of the Hilbert scheme. Collect. Math. 64, 39–59 (2013). https://doi.org/10.1007/s13348-011-0055-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-011-0055-7