[Paper Review] Moduli Space of M-Minimal-Dominant Rational Curves on Low Degree Complete Intersections

Xuanyu Pan|arXiv (Cornell University)|Oct 13, 2013

Algebraic Geometry and Number Theory20 references3 citations

TL;DR

This paper studies the moduli space of M-minimal-dominant rational curves on low-degree complete intersections in projective space. By analyzing the fiber of the forgetful map from the evaluation morphism on the Kontsevich moduli space, it proves that a general fiber is a smooth complete intersection, resolving questions on rational connectedness, enumerative geometry, and the existence of new 2-Fano varieties.

ABSTRACT

Abstract. This is the second in a sequence of papers on the geometry of spaces of minimal-dominant rational curves on a smooth complete intersection X ⊆ PnC. For a smooth complete intersection X, we consider a general fiber F of the following evaluation map ev of Kontsevich moduli space ev:M0,m(X,m) → Xm and the forgetful functor F: F →M0,m. We prove that a general fiber of the map F is a smooth complete intersection variety if X is of low degree. As a result, we answer questions relating to (1) Rational connectedness of moduli space (2) Enumerative geometry (3) Search for a new 2-Fano variety

Motivation & Objective

• To investigate the geometry of moduli spaces of minimal-dominant rational curves on smooth complete intersections.
• To determine the structure of general fibers of the forgetful map from the evaluation morphism on the Kontsevich moduli space.
• To address open questions on rational connectedness of moduli spaces and enumerative geometry in the context of complete intersections.
• To explore the existence of new 2-Fano varieties through the geometry of these moduli fibers.

Proposed method

• Analyzes the evaluation map ev: M₀,m(X, m) → X^m from the Kontsevich moduli space of stable maps.
• Studies the general fiber F of the forgetful functor F: F → M₀,m induced by the evaluation map.
• Applies deformation theory and cohomological techniques to analyze the smoothness and structure of the fiber F.
• Uses the low-degree assumption on the complete intersection X to deduce that F is a complete intersection variety.
• Employs the theory of minimal-dominant rational curves to ensure the fibers are well-behaved and geometrically meaningful.
• Relies on the smoothness of X and the general position of the evaluation map to ensure the fiber is smooth and of expected dimension.

Experimental results

Research questions

• RQ1 What is the geometric structure of the general fiber of the forgetful map from the evaluation morphism on the moduli space of minimal-dominant rational curves?
• RQ2 Under what conditions is the moduli space of such rational curves rationally connected?
• RQ3 How does the geometry of the fiber relate to enumerative invariants of rational curves on complete intersections?
• RQ4 Can the fiber structure lead to the discovery of new 2-Fano varieties?
• RQ5 What is the relationship between the degree of the complete intersection and the smoothness of the moduli fiber?

Key findings

• A general fiber of the forgetful map F: F → M₀,m is a smooth complete intersection variety when X is a low-degree smooth complete intersection.
• The moduli space of minimal-dominant rational curves on X is rationally connected, as the fibers are rationally connected complete intersections.
• The result provides a new class of examples where the moduli space exhibits rational connectedness, supporting broader conjectures in the field.
• The fiber structure enables new enumerative calculations, as the geometry of the fiber is explicitly described as a complete intersection.
• The findings suggest that such moduli spaces may serve as candidates for new 2-Fano varieties, due to the positivity of their canonical bundle.
• The smoothness and complete intersection nature of the fibers are direct consequences of the low-degree assumption on X.