Geometric Manin's Conjecture and rational curves

Abstract

Let $X$ be a smooth projective Fano variety over the complex numbers. We study the moduli spaces 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.

arXiv comment29 pages; a minor gap in Section 7 has been addressed. To appear in Compositio Mathematica

References in corpus (10)

• Rational points on some Fano cubic bundles
• Produit eulérien motivique et courbes rationnelles sur les variétés toriques
• On the geometry of thin exceptional sets in Manin's Conjecture
• One Cycles on Rationally Connected Varieties
• Rational curves on smooth hypersurfaces of low degree
• Rational curves and instantons on the Fano threefold $Y_5$
• Spaces of rational curves in complete intersections
• Cylinders in del Pezzo surfaces
• On Fujita invariants of subvarieties of a uniruled variety
• Asymptotic behaviour of rational curves

Cited by in corpus (14)

• Manin's Conjecture and the Fujita invariant of finite covers
• A desingularization of Kontsevich's compactification of twisted cubics in $V_5$
• Classifying sections of del Pezzo fibrations, II
• Rational curves on del Pezzo surfaces in positive characteristic
• Fano mirror periods from the Frobenius structure conjecture
• Manin's b-constant in families
• Rational curves on $V_5$ and rational simple connectedness
• An introduction to Geometric Manin's conjecture
• On upper bounds of Manin type
• On the asymptotic enumerativity property for Fano manifolds
• Rational curves on prime Fano threefolds of index 1
• Movable Bend and Break for sections of del Pezzo fibrations
• The geometric exceptional set in Manin's conjecture for Batyrev and Tschinkel's example
• Motivic distribution of rational curves and twisted products of toric varieties