Leading terms of generalized Plücker formulas (2404.18859v2)
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.