Interpolation for Brill–Noether curves in \({\mathbb {P}}^4\)


We compute the number of general points through which a general Brill–Noether curve in \({\mathbb {P}}^4\) passes. We also prove an analogous theorem when some points are constrained to lie in a transverse hyperplane. As explained in [10], these results play an essential role in the first author’s proof of the Maximal Rank Conjecture [9].

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  1. 1.

    Atanasov, A., Larson, E., Yang, D.: Interpolation for Normal Bundles of General Curves. Memoirs of the American Mathematical Society, vol. 257(1234). American Mathematical Society, Providence (2018).

  2. 2.

    Atanasov, A.V.: Interpolation and Vector Bundles on Curves. PhD Thesis, Harvard University (2015)

  3. 3.

    Ballico, E.: An interpolation problem for the normal bundle of curves of genus \(g\ge 2\) and high degree in \(\mathbb{P}^r\). Commun. Algebra 45(2), 822–827 (2017)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Hartshorne, R., Hirschowitz, A.: Smoothing algebraic space curves. In: Casas-Alvero, E., Welters, G.E., Xambó-Descamps, S. (eds.) Algebraic Geometry, Sitges. Lecture Notes in Mathematics, vol. 1124, pp. 98–131. Springer, Berlin (1985)

    Google Scholar 

  5. 5.

    Hirschowitz, A.: La méthode d’Horace pour l’interpolation à plusieurs variables. Manuscripta Math. 50, 337–388 (1985)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Iliev, H.: On the irreducibility of the Hilbert scheme of space curves. Proc. Amer. Math. Soc. 134(10), 2823–2832 (2006)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Larson, E.: Constructing reducible Brill–Noether curves (2016). arXiv:1603.02301

  8. 8.

    Larson, E.: The generality of a section of a curve (2016). arXiv:1605.06185

  9. 9.

    Larson, E.: The maximal rank conjecture (2017). arXiv:1711.04906

  10. 10.

    Larson, E.: Degenerations of curves in projective space and the maximal rank conjecture (2018). arXiv:1809.05980

  11. 11.

    Perrin, D.: Courbes passant par \(k\) points généraux de \({\mathbb{P}}^3\). C. R. Acad. Sci. Paris Sér. I Math. 299(10), 451–453 (1984)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Stevens, J.: On the number of points determining a canonical curve. Nederl. Akad. Wetensch. Indag. Math. 51(4), 485–494 (1989)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Vogt, I.: Interpolation for Brill-Noether space curves. Manuscripta Math. 156(1–2), 137–147 (2018)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Waring, E.: Problems concerning interpolations. Philos. Trans. R. Soc. 69, 59–67 (1779)

    Article  Google Scholar 

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We would like to thank Joe Harris for his guidance, as well as Ravi Vakil, Izzet Coskun, and members of the MIT and Harvard math departments for helpful conversations.

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Correspondence to Isabel Vogt.

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The first author acknowledges the generous support both of the Fannie and John Hertz Foundation, and of the Department of Defense (NDSEG fellowship). The second author acknowledges the generous support of the National Science Foundation Graduate Research Fellowship Program under DGE-1122374.

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Larson, E., Vogt, I. Interpolation for Brill–Noether curves in \({\mathbb {P}}^4\). European Journal of Mathematics (2020).

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