Sequences of Enumerative Geometry: Congruences and Asymptotics

We study the integer sequence υn of numbers of lines in hypersurfaces of degree 2 n − 3 of ℙ n , n > 1. We prove a number of congruence properties of these numbers of several different types. Furthermore, the asymptotics of the υ n are described (in an appendix by Don Zagier). Finally, an attempt is made at carrying out a similar analysis for numbers of rational plane curves.

[1]  Laurent Manivel,et al.  Symmetric Functions Schubert Polynomials and Degeneracy Loci , 2001 .

[2]  D. R. Heath-Brown,et al.  An Introduction to the Theory of Numbers, Sixth Edition , 2008 .

[3]  Jan Stienstra,et al.  On the Picard-Fuchs equation and the formal brauer group of certain ellipticK3-surfaces , 1985 .

[4]  S. Yau,et al.  Arithmetic properties of mirror map and quantum coupling , 1994, hep-th/9411234.

[5]  S. Yau.  A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory , 1998 .

[6]  Isaac Leroy Hines.  Symmetric functions , 2021, Tau Functions and their Applications.

[7]  Y. Moshe.  The density of 0's in recurrence double sequences , 2003 .

[8]  E. D. Giorgi.  Selected Papers , 2006 .

[9]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[10]  D. Zagier.  Vassiliev invariants and a strange identity related to the Dedekind eta-function , 2001 .

[11]  Serge Tabachnikov,et al.  Arithmetical properties of binomial coefficients , 2007 .

[12]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[13]  B. L. Waerden.  Zur algebraischen Geometrie , 1937 .

[14]  On some arithmetical properties of middle binomial coefficients , 1998 .

[15]  Yossi Moshe.  The distribution of elements in automatic double sequences , 2005, Discret. Math..

[16]  M. Kontsevich,et al.  Gromov-Witten classes, quantum cohomology, and enumerative geometry , 1994 .

[17]  Xenia de la Ossa,et al.  A Pair of Calabi-Yau manifolds as an exactly soluble superconformal theory , 1991 .

[18]  Johan P. Hansen,et al.  INTERSECTION THEORY , 2011 .