[Paper Review] Rational families of vector bundles on curves, I

Ana-Maria Castravet|ArXiv.org|Feb 12, 2003

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper studies rational curves on the moduli space M of rank 2 stable vector bundles with fixed determinant of degree 1 on a genus g ≥ 2 curve C. Using extensions of line bundles and skyscraper sheaves, it identifies two irreducible components in the space of rational curves of degree k: the 'nice' component (where general curves are very free for large k) and the 'almost nice' component (where general curves are free with specific tangent bundle splitting). The maximal rationally connected fibration of both components is the Jacobian J(C).

ABSTRACT

Let C be a smooth complex projective curve of genus at least 2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1. For any k>1, we find two irreducible components of the space of rational curves of degree k on M. One component, which we call the nice component has the property that the general element is a very free curve if k is sufficiently large. The other component has the general element a free curve. Both components have the expected dimension and their maximal rationally connected fibration is the Jacobian of the curve C.

Motivation & Objective

• To classify irreducible components of the space of rational curves of degree k on the moduli space M of rank 2 stable vector bundles with fixed determinant of degree 1 on a genus g ≥ 2 curve.
• To determine the dimension and geometric properties (e.g., freeness, very freeness) of these components.
• To identify the maximal rationally connected (MRC) fibration of each component and show it is the Jacobian J(C).
• To establish that for large k, the general curve in the 'nice' component is very free, indicating strong rational connectivity.

Proposed method

• Constructs rational curves via extensions of line bundles: for odd k = 2e + 1, uses extensions 0 → L → E → L⁻¹⊗ξ → 0 with deg(L) = -e.
• For even k, uses extensions 0 → E → E′ → O_D → 0 with D ∈ Sym^e(C), where E has det(ξ(-D)).
• Analyzes the associated rational maps from projective spaces P(V_L) or P(V_{D,E}) to M, showing they are defined outside codimension ≥2 loci.
• Applies deformation theory and the base-point-free pencil trick to prove surjectivity of determinant maps, ensuring the existence of desired rational curves.
• Uses the fact that H¹(L⊗M⁻¹(-p)) = 0 for carefully chosen line bundles to ensure surjectivity of the determinant map in the even degree case.
• Deforms direct sums of line bundles to stable bundles to prove the general case via specialization and openness of the property.

Experimental results

Research questions

• RQ1 What are the irreducible components of the space Hom_k(P¹, M) for rational curves of degree k on the moduli space M?
• RQ2 What is the dimension of these components, and do they have the expected dimension 2k + 3g - 3?
• RQ3 Under what conditions is the general rational curve in each component free or very free?
• RQ4 What is the maximal rationally connected (MRC) fibration of each component?
• RQ5 How does the geometry of the curve C, particularly its Jacobian J(C), relate to the structure of rational curves on M?

Key findings

• For any k ≥ 1, there exists an irreducible component M of Hom_k(P¹, M) of the expected dimension 2k + 3g - 3, which is unobstructed in the general point.
• The MRC fibration of the 'nice' component M is a rational map M ⇢ J(C), which is dominant for all k except k = 2 in the genus 2 case.
• For large k, the general rational curve in the 'nice' component is very free, indicating strong rational connectivity in that component.
• The 'almost nice' component consists of morphisms f: P¹ → M that are k-to-1 onto a line in M, with f*TM ≅ O^g ⊕ A for some positive bundle A on P¹.
• In the genus 2 case, M is a complete intersection of two quadrics in P⁵, and both components exist for all k ≥ 1, with the same MRC fibration to J(C).
• The bound for very freeness in the even degree 'nice' component is improved beyond the initial 6g - 6 via deformation of reducible curves from odd-degree free curves.