Geometric Manin’s conjecture and rational curves

By Brian Lehmann, Sho Tanimoto. Compositio Mathematica, 2017

56 citations recorded by the source corpus.

Let $X$ be a smooth projective Fano variety over the complex numbers. We study the moduli space of rational curves on $X$ using the perspective of Manin’s conjecture. In particular, we bound the dimension and number of components of spaces of rational curves on $X$ . We propose a geometric Manin’s conjecture predicting the growth rate of a counting function associated to the irreducible components of these moduli spaces.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8f87e6c37fbc2fbd0532d0f351f4baee275fe71e

Citations in the corpus

• Geometric consistency of Manin's conjecture (2018, unknown): “However, based on the properties of rational curves on low degree hypersurfaces proved by [HRS04], [BK13], [RY19], [BV17] and the connection with a and b invariants proved in [LT19], we expect that Z1 and Z2 are empty for general”
• Geometric (2019, unknown): “Classification constitutes a core cognitive challenge for both biological and artificial intelligence systems, with many tasks potentially reducible to classification problems (Cover, 1965; Chung et al., 2018; DiCarlo et al., 2012; Bernardi et al., 2020).”