Leading terms of generalized Plücker formulas
Abstract: Generalized Pl\"ucker numbers are defined to count certain types of tangent lines of generic degree $d$ complex projective hypersurfaces. They can be computed by identifying them as coefficients of GL(2)-equivariant cohomology classes of certain invariant subspaces, the so-called coincident root strata, of the vector space of homogeneous degree $d$ complex polynomials in two variables. In an earlier paper L\'aszl\'o M. Feh\'er and the author gave a new, recursive method for calculating these classes. Using this method, we showed that -- similarly to the classical Pl\"ucker formulas counting the bitangents and flex lines of a degree $d$ plane curve -- generalized Pl\"ucker numbers are polynomials in the degree $d$. In this paper, by further analyzing our recursive formula, we determine the leading terms of all the generalized Pl\"ucker formulas.
- S. J. Colley. Lines having specified contact with projective varieties. In Proceedings of the 1984 Vancouver Conference in Algebraic Geometry, Annual seminar of the Canadian Mathematical Society, pages 47–70. American Mathematical Society, 1986.
- Plücker formulas using equivariant cohomology of coincident root strata, 2023. https://arxiv.org/pdf/2312.06430.pdf.
- Coincident root loci of binary forms. Michigan Math. J., 54(2):375–392, 2006.
- M. Kazarian. Morin maps and their characteristic classes. https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.486.8275&rep=rep1&type=pdf, 2006.
- B. Kőműves. Thom polynomials via restriction equations. Master’s thesis, Eötvös Lóránt University Budapest, 2003.
- F. Kirwan. Cohomology of quotients in symplectic and algebraic geometry. Number 31 in Mathematical Notes. Princeton UP, 1984.
- S. L. Kleiman. The enumerative theory of singularities. Uspekhi Matematicheskikh Nauk, 1977.
- S. L. Kleiman. Multiple-point formulas I: Iteration. Acta Mathematica, 147:13 – 49, 1981.
- S. L. Kleiman. Multiple Point Formulas for Maps, pages 237–252. Birkhäuser Boston, Boston, MA, 1982.
- Patrick Le Barz. Formules multisécantes pour les courbes gauches quelconques. In Enumerative geometry and classical algebraic geometry (Nice, 1981), volume 24 of Progr. Math., pages 165–197. Birkhäuser, Boston, Mass., 1982.
- PGL2𝑃𝐺subscript𝐿2PGL_{2}italic_P italic_G italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-equivariant strata of point configurations in ℙ1superscriptℙ1\mathbb{P}^{1}blackboard_P start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 23(2):569–621, 2022.
- Burt Totaro. The Chow ring of a classifying space. In Algebraic K𝐾Kitalic_K-theory (Seattle, WA, 1997), volume 67 of Proc. Sympos. Pure Math., pages 249–281. Amer. Math. Soc., Providence, RI, 1999.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.