Fields of definition of curves of a given degree

2019/01/31 by David Holmes, Holmes, David, Nick Rome +1
Computer Science · Engineering · Mathematics · #11N36 #11P55 #14G05 #14H10 #14J70 #14N35 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1901.11294

openalex publication_date 2019/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Kontsevich and Manin gave a formula for the number N_e of rational plane curves of degree e through 3e-1 points in general position in the plane. When these 3e-1 points have coordinates in the rational numbers, the corresponding set of N_e rational curves has a natural Galois-module structure. We make some extremely preliminary investigations into this Galois module structure, and relate this to the deck transformations of the generic fibre of the product of the evaluation maps on the moduli space of maps. We then study the asymptotics of the number of rational points on hypersurfaces of low degree, and use this to generalise our results by replacing the projective plane by such a hypersurface.

Citations

• Enumeration of n-fold tangent hyperplanes to a surface

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