[Paper Review] Bundles of rank 2 with small Clifford index on algebraic curves

Herbert Lange, P. E. Newstead|arXiv (Cornell University)|May 22, 2011

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper constructs stable rank-2 vector bundles on algebraic curves of genus $ g \geq 12 $ with maximal classical Clifford index $ \left[\frac{g-1}{2}\right] $, achieving the minimal possible rank-2 Clifford index $ \frac{1}{2}\left[\frac{g-1}{2}\right] + 2 $. The construction uses curves on K3 surfaces with Picard number 2, controlling $(-2)$-curves to ensure the desired stability and Clifford index bounds, thereby providing sharp counterexamples to Mercat's conjecture for rank 2 bundles.

ABSTRACT

In this paper, we remove a technical hypothesis from a previous paper. This allows us to construct stable bundles of rank 2 with Clifford index significantly smaller than the classical Clifford index on suitably chosen curves of any genus $\ge13$.

Motivation & Objective

• To construct stable rank-2 vector bundles on algebraic curves of genus $ g \geq 12 $ with maximal classical Clifford index $ \left[\frac{g-1}{2}\right] $.
• To achieve the minimal possible value of the rank-2 Clifford index $ \operatorname{Cliff}_2(C) $ for such curves.
• To improve upon earlier results by weakening the hypotheses in [8, Theorem 1.1], showing the new conditions are best possible.
• To demonstrate that the rank-2 Clifford index can be strictly less than the classical Clifford index, thus providing sharp counterexamples to Mercat's conjecture for rank 2.

Proposed method

• Constructing curves $ C $ of genus $ g \geq 12 $ as complete intersections on a smooth $ (2,3) $-type K3 surface $ S \subset \mathbb{P}^4 $ with $ \operatorname{Pic}(S) = \mathbb{Z}H \oplus \mathbb{Z}C $.
• Using the geometry of $ S $, particularly the absence or controlled presence of $(-2)$-curves, to ensure the stability and desired cohomological properties of the bundle $ E $.
• Defining the rank-2 Clifford index via $ \gamma(E) = \mu(E) - 2\frac{h^0(E)}{2} + 2 $, and minimizing this over semistable bundles with $ h^0(E) \geq 4 $ and $ \mu(E) \leq g-1 $.
• Analyzing the intersection theory on $ S $, especially for divisors $ D \sim mH + nC $, to bound $ h^0(C, \mathcal{O}_C(D)) $ and ensure $ \gamma(E) = \frac{g-s}{2} - 2 $ for $ d = g - s $.
• Proving that the minimal value of $ \operatorname{Cliff}_2(C) $ is $ \frac{1}{2}\left[\frac{g-1}{2}\right] + 2 $ by showing that $ f(m,n) \geq \left[\frac{g-1}{2}\right] $ for all relevant $ (m,n) $, with equality cases controlled.
• Using the fact that $ H|_C $ computes the classical Clifford index when $ g = 2s + 14 $ or $ g = 2s + 15 $, ensuring $ \operatorname{Cliff}(C) = \left[\frac{g-1}{2}\right] $.

Experimental results

Research questions

• RQ1 Can the rank-2 Clifford index $ \operatorname{Cliff}_2(C) $ be strictly less than the classical Clifford index $ \operatorname{Cliff}(C) $ for curves of maximal Clifford index?
• RQ2 Is the bound $ \operatorname{Cliff}_2(C) = \frac{1}{2}\left[\frac{g-1}{2}\right] + 2 $ the minimal possible value for curves with $ \operatorname{Cliff}(C) = \left[\frac{g-1}{2}\right] $?
• RQ3 Are the hypotheses in Theorem 3.3, requiring $ g \geq 2s + 14 $, best possible, or does the result fail for $ g = 2s + 13 $?
• RQ4 Does there exist a curve $ C $ with $ \operatorname{Cliff}(C) = \gamma $ and $ \operatorname{Cliff}_2(C) = \gamma' $ for any $ \frac{\gamma}{2} + 2 < \gamma' < \gamma $?
• RQ5 Is $ \operatorname{Cliff}_2(C) = \operatorname{Cliff}(C) $ true for the general curve $ C $ of any genus, as conjectured by Mercat?

Key findings

• For any genus $ g \geq 12 $, there exists a curve $ C $ with $ \operatorname{Cliff}(C) = \left[\frac{g-1}{2}\right] $ and a stable rank-2 bundle $ E $ of degree $ d = g - s $ such that $ \gamma(E) = \frac{g-s}{2} - 2 $, implying $ \operatorname{Cliff}_2(C) \leq \frac{g-s}{2} - 2 $.
• The minimal possible value of $ \operatorname{Cliff}_2(C) $ for curves with $ \operatorname{Cliff}(C) = \left[\frac{g-1}{2}\right] $ is $ \frac{1}{2}\left[\frac{g-1}{2}\right] + 2 $, and this bound is achieved.
• The result is sharp: for $ g = 2s + 13 $, the construction fails because $ \gamma(E) < \frac{1}{2}\left[\frac{g-1}{2}\right] + 2 $, contradicting known bounds.
• For any integer $ \gamma \geq 5 $, there exists a curve $ C $ with $ \operatorname{Cliff}(C) = \gamma $ and $ \operatorname{Cliff}_2(C) = \frac{\gamma}{2} + 2 $, realizable with genus $ 2\gamma + 1 $ or $ 2\gamma + 2 $.
• The construction relies on curves on K3 surfaces with Picard number 2, and the presence of $(-2)$-curves is controlled to avoid disrupting the cohomological estimates.
• The result shows that Mercat's conjecture fails for rank-2 bundles on curves of maximal Clifford index, and the failure is quantitatively minimal.