Hyperflex loci of hypersurfaces

• Bertone, Cristina ^[1] ; Weimann, Martin ^[2]
  1. [1] University of Turin

     University of Turin

     Torino, Italia

     
  2. [2] Université de Caen Basse-Normandie

     Université de Caen Basse-Normandie

     Arrondissement de Caen, Francia

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 77, Fasc. 3, 2026, págs. 623-641
• Idioma: inglés
• DOI: 10.1007/s13348-025-00484-w
• Texto completo no disponible (Saber más ...)
• Resumen

  • The k-flex locus of a projective hypersurface V\subset \mathbb {P}^n is the locus of points p\in V such that there is a line with contact order at least k with V at p. Unexpected contact orders occur when k\ge n+1. The case k=n+1 is known as the classical flex locus, which has been studied in details in the literature. This paper is dedicated to computing the dimension and the degree of the k-flex locus of a general degree d hypersurface for any value of k. As a corollary, we compute the dimension and the degree of the biggest ruled subvariety of a general hypersurface. We show moreover that through a generic k-flex point, there passes a unique k-flex line which has contact order exactly k if k\le d. The proof is based on the computation of the top Chern class of a certain vector bundle of relative principal parts, inspired by and generalizing a work of Eisenbud and Harris.

• Referencias bibliográficas
  • Busé, L., D’Andrea, C., Sombra, M., Weimann, M.: The geometry of the flex locus of a hypersurface. Pacific J. Math. 304(2), 419–437 (2020)
  • Eisenbud, D., Harris, J.: 3264 and All That; Intersection Theory in Algebraic Geometry. Cambridge University (2016)
  • Fulton, W.: Intersection Theory. Springer, New York (1998)
  • Guth, L., Katz, N.H.: On the Erdös distinct distances problem in the plane. Ann. Math. 181(2), 155–190 (2015)
  • Hartshorne, R.: Algebraic geometry Graduate Texts in Mathematics. Springer, New York-Heidelberg (1977)
  • Katz, N.: The flecnode polynomial: a central object in incidence geometry Proc. ICM 2014, Vol. III, 303–314 (2014)
  • Kollár, J.: Szemerédi-Trotter-type theorems in dimension 3. Adv. Math. 271, 30–61 (2015)
  • Reuveni, S.: Catalan’s trapezoids Probab. Engrg. Inform. Sci. 28(3), 353–361 (2014)
  • Salmon, G.: On quaternary cubics. Phil. Trans. R. Soc. London 150, 229–239 (1860)
  • Sharir, M., Solomon, N.: Incidences between points and lines on two- and three- dimensional varieties. Discrete Comput. Geom. 59, 88–130 (2018)
  • Tao, T.: The Monge-Cayley-Salmon theorem via classical differential geometry., https://terrytao.wordpress.com/2014/03/28/, (2014)
  • Thomas, K.: Catalan Numbers with Applications. Oxford University Press, USA (2008)