[Paper Review] A stratification of the moduli space of vector bundles on curves

L. Brambila‐Paz, Herbert Lange|arXiv (Cornell University)|Aug 18, 1997

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper introduces a stratification of the moduli space of stable vector bundles on a smooth projective curve by defining invariants $ s_k(E) = k\cdot \deg E - r\cdot \max \deg F $, where $ F $ ranges over subbundles of rank $ k $. It proves that for $ g \geq \frac{r+1}{2} $, the stratum $ \mathcal{M}^0(r,d,k,s) $ is non-empty and computes its dimension explicitly, showing it depends on whether $ s $ is below or above $ k(r-k)(g-1) $.

ABSTRACT

Let $E$ be a vector bundle of rank $r\geq 2$ on a smooth projective curve $C$ of genus $g \geq 2$ over an algebraically closed field $K$ of arbitrary characteristic. For any integer with $1\le k\le r-1$ we define $${\se}_k(E):=k°E-r\max°F.$$ where the maximum is taken over all subbundles $F$ of rank $k$ of $E$. The ${s}_k$ gives a stratification of the moduli space ${\cal M}(r,d)$ of stable vector bundles of rank $r$ and degree on $d$ on $C$ into locally closed subsets ${\calM}(r,d,k,s)$ according to the value of $s$ and $k$. There is a component ${\cal M}^0(r,d,k,s)$ of ${\cal M}(r,d,k,s)$ distinguish by the fact that a general $E\in {\cal M}^0(r,d,k,s)$ admits a stable subbundle $F$ such that $E/F$ is also stable. We prove: {\it For $g\ge \frac{r+1}{2}$ and $0

Motivation & Objective

• To understand the structure of the moduli space $ \mathcal{M}(r,d) $ of stable vector bundles on a smooth projective curve of genus $ g \geq 2 $.
• To define and analyze a new stratification of $ \mathcal{M}(r,d) $ based on the invariant $ s_k(E) $, measuring the maximality of subbundle degrees.
• To determine the non-emptiness and dimension of the strata $ \mathcal{M}^0(r,d,k,s) $, where a general bundle admits a stable subbundle with stable quotient.
• To establish precise dimension formulas for these strata depending on the value of $ s $ relative to $ k(r-k)(g-1) $.

Proposed method

• Define $ s_k(E) = k \cdot \deg E - r \cdot \max \deg F $, where $ F $ runs over all rank-$ k $ subbundles of $ E $.
• Use the invariant $ s_k(E) $ to stratify $ \mathcal{M}(r,d) $ into locally closed subsets $ \mathcal{M}(r,d,k,s) $.
• Focus on the open component $ \mathcal{M}^0(r,d,k,s) $, consisting of bundles $ E $ that admit a stable subbundle $ F $ of rank $ k $ such that $ E/F $ is also stable.
• Apply techniques from algebraic geometry and vector bundle theory, particularly over algebraically closed fields of arbitrary characteristic.
• Use dimension theory and stability conditions to compute the dimension of $ \mathcal{M}^0(r,d,k,s) $ in two cases: $ s < k(r-k)(g-1) $ and $ s \geq k(r-k)(g-1) $.
• Derive the dimension formula using the rank and degree of the bundles and the genus $ g $ of the curve.

Experimental results

Research questions

• RQ1 For which values of $ s $ is the stratum $ \mathcal{M}^0(r,d,k,s) $ non-empty in the moduli space $ \mathcal{M}(r,d) $?
• RQ2 What is the dimension of the stratum $ \mathcal{M}^0(r,d,k,s) $, and how does it depend on the value of $ s $ relative to $ k(r-k)(g-1) $?
• RQ3 Under what conditions does a general bundle in $ \mathcal{M}^0(r,d,k,s) $ admit a stable subbundle with stable quotient?
• RQ4 How does the invariant $ s_k(E) $ capture the geometric structure of the moduli space of vector bundles on curves?

Key findings

• For $ g \geq \frac{r+1}{2} $ and $ 0 < s \leq k(r-k)(g-1) + (r+1) $, the stratum $ \mathcal{M}^0(r,d,k,s) $ is non-empty.
• When $ s < k(r-k)(g-1) $, the dimension of $ \mathcal{M}^0(r,d,k,s) $ is $ (r^2 + k^2 - rk)(g-1) + s - 1 $.
• When $ s \geq k(r-k)(g-1) $, the dimension of $ \mathcal{M}^0(r,d,k,s) $ is $ r^2(g-1) + 1 $.
• The stratum $ \mathcal{M}^0(r,d,k,s) $ is characterized by the existence of a stable subbundle $ F $ of rank $ k $ such that $ E/F $ is also stable.
• The invariant $ s_k(E) $ provides a well-defined stratification of $ \mathcal{M}(r,d) $ into locally closed subsets indexed by $ k $ and $ s $.
• The dimension formulas are valid under the condition $ s \equiv kd \mod r $, ensuring integrality of the invariant.