# Sequences of Enumerative Geometry: Congruences and Asymptotics

@article{Grnberg2008SequencesOE, title={Sequences of Enumerative Geometry: Congruences and Asymptotics}, author={Daniel B. Gr{\"u}nberg and Pieter Moree and Don Zagier}, journal={Experimental Mathematics}, year={2008}, volume={17}, pages={409 - 426} }

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.

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## References

SHOWING 1-10 OF 42 REFERENCES

### Gromov-Witten classes, quantum cohomology, and enumerative geometry

- Mathematics
- 1994

The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic…

### Quantum intersection rings

- Mathematics
- 1994

Within the broadly defined subject of topological field theory E. Witten suggested in [1] to study generalized “intersection numbers” on a compactified moduli space \({\bar M_{g,n}}\) of Riemann…

### INTERSECTION THEORY

- Mathematics
- 2007

I provide more details to the intersection theoretic results in [1]. CONTENTS 1. Transversality and tubular neighborhoods 1 2. The Poincaré dual of a submanifold 4 3. Smooth cycles and their…

### Arithmetic properties of mirror map and quantum coupling

- Mathematics
- 1996

We study some arithmetic properties of the mirror maps and the quantum Yukawa couplings for some 1-parameter deformations of Calabi-Yau manifolds. First we use the Schwarzian differential equation,…

### Symmetric Functions Schubert Polynomials and Degeneracy Loci

- Mathematics
- 2001

Introduction The ring of symmetric functions Schubert polynomials Schubert varieties A brief introduction to singular homology Bibliography Index.

### An Introduction to the Theory of Numbers

- MathematicsNature
- 1946

THIS book must be welcomed most warmly into X the select class of Oxford books on pure mathematics which have reached a second edition. It obviously appeals to a large class of mathematical readers.…

### An Introduction to the Theory of Numbers, Sixth Edition

- Psychology
- 2008

Read more and get great! That's what the book enPDFd an introduction to the theory of numbers 5th edition will give for every reader to read this book. This is an on-line book provided in this…