The matroid stratification of the Hilbert scheme of points on \(\mathbb {P}^1\)

Abstract

Given a homogeneous ideal I in a polynomial ring over a field, one may record, for each degree d and for each polynomial \(f\in I_d\), the set of monomials in f with nonzero coefficients. These data collectively form the tropicalization of I. Tropicalizing ideals induces a “matroid stratification” on any (multigraded) Hilbert scheme. Very little is known about the structure of these stratifications. In this paper, we explore many examples of matroid strata, including some with interesting combinatorial structure, and give a convenient way of visualizing them. We show that the matroid stratification in the Hilbert scheme of points \((\mathbb {P}^1)^{[k]}\) is generated by all Schur polynomials in k variables. We end with an application to the T-graph problem of \((\mathbb {A}^2)^{[n]};\) classifying this graph is a longstanding open problem, and we establish the existence of an infinite class of edges.

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Notes

  1. 1.

    There is a more general notion of multigrading that we will not need, see [7].

  2. 2.

    Positivity of \(\mathbf {a}\) is necessary here; otherwise \(R_d/(I\cap R_d)\) need not be finite-dimensional.

References

  1. 1.

    Altmann, K., Sturmfels, B.: The graph of monomial ideals. J. Pure Appl. Algebra 201(1), 250–263 (2005)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Conca, A., Krattenthaler, C., Watanabe, J.: Regular sequences of symmetric polynomials. Rendiconti del Seminario Matematico della Università di Padova 121, 179–199 (2009)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Evain, L.: Incidence relations among the Schubert cells of equivariant punctual Hilbert schemes. Math. Z. 242(4), 743–759 (2002)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Evain, L.: Irreducible components of the equivariant punctual Hilbert schemes. Adv. Math. 185(2), 328–346 (2004)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Fink, A., Giansiracusa, J., Giansiracusa, N.: Projective hypersurfaces in tropical scheme theory. (in preparation)

  6. 6.

    Hering, M., Maclagan, D.: The \(T\)-graph of a multigraded Hilbert scheme. Exp. Math. 21(3), 280–297 (2012)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Haiman, M., Sturmfels, B.: Multigraded Hilbert schemes. J. Algebr. Geom. 13, 725–769 (2004)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Iarrobino, A.: Punctual Hilbert schemes. Bull. Am. Math. Soc. 78(5), 819–823 (1972)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Maclagan, D.: TEdges: Package for Macaulay2. http://homepages.warwick.ac.uk/staff/D.Maclagan/papers/TEdgesFiles/TEdges.m2 (2009)

  10. 10.

    Maclagan, D., Rincón, F.: Tropical ideals. Compos. Math. 154(3), 640–670 (2018)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Maclagan, D., Smith, G.: Smooth and irreducible multigraded Hilbert schemes. Adv. Math. 223(5), 1608–1631 (2010)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Okounkov, A., Pandharipande, R.: Quantum cohomology of the Hilbert scheme of points in the plane. Invent. Math. 179(3), 523–557 (2010)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Oxley, J.: Matroid Theory. Oxford University Press, Oxford (2006)

    Google Scholar 

  14. 14.

    Zajaczkowska, M.A.: Tropical Ideals with Hilbert Function Two. Ph.d. thesis, University of Warwick (2018)

Download references

Acknowledgements

I am grateful to Diane Maclagan, for guidance and information about the T-graph problem, for pointing out that the sequences of matroids studied in this paper are a special case of tropical ideals, for referring me to known results in the field [5, 10, 14], for giving feedback, and for writing the useful and convenient TEdges package for Macaulay2. I am also grateful to Tim Ryan—we jointly generated data that led to Corollary 5.11. I would also like to thank Noah Giansiracusa for introducing me to [5], Rohini Ramadas for helpful conversations about tropical geometry, and the anonymous referee for suggesting many improvements. This project was supported by NSF DMS-1645877, and by postdoctoral positions at the Simons Center for Geometry and Physics and at Northeastern University.

Funding

Rob Silversmith: National Science Foundation grant DMS-1645877, Simons Center for Geometry and Physics, and Northeastern University.

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Silversmith, R. The matroid stratification of the Hilbert scheme of points on \(\mathbb {P}^1\). manuscripta math. (2021). https://doi.org/10.1007/s00229-021-01280-z

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Keywords

  • Tropical ideals
  • Multigraded Hilbert schemes
  • Schur polynomials

Mathematics Subject Classification

  • 14C05
  • 05E99