[Paper Review] Rank two globally generated vector bundles with c_1 \leq 5

Ludovica Chiodera, Philippe Ellia|arXiv (Cornell University)|Nov 24, 2011

Algebraic Geometry and Number Theory10 references18 citations

TL;DR

This paper classifies globally generated rank two vector bundles on projective spaces $\mathbb{P}^n$ for $n \geq 3$ and $c_1 \leq 5$. It shows that for $n \geq 4$, all such bundles split as direct sums of line bundles. For $n = 3$, indecomposable bundles exist only for specific Chern classes $c_1 \in \{2,4,5\}$, $c_2 \in \{2,5,6,7,8,8,10,12\}$, with the case $(c_1,c_2) = (5,12)$ remaining open. The classification relies on linking theory and cohomological constraints via liaison and stability conditions.

ABSTRACT

We classify rank two globally generated vector bundles on P^n, n > 2, with c_1 \leq 5. The classification is complete but for one case (n = 3, c_1 = 5, c_2 = 12)

Motivation & Objective

• To classify globally generated rank two vector bundles on $\mathbb{P}^n$ for $n \geq 3$ and $c_1 \leq 5$.
• To determine which such bundles are indecomposable and under what conditions they exist.
• To resolve the existence question for the case $(c_1, c_2) = (5, 12)$ in $\mathbb{P}^3$, which remains open.
• To establish a complete classification for $n \geq 4$, showing all such bundles must split.
• To characterize the associated curves of zero-locus of general sections via liaison theory and canonical conditions.

Proposed method

• Use of the exact sequence $0 \to \mathcal{O} \to E \to \mathcal{I}_C(c_1) \to 0$ for indecomposable $E$, where $C$ is a smooth curve with $\omega_C(4 - c_1) \simeq \mathcal{O}_C$.
• Application of liaison theory to link the zero-locus curve $C$ to another curve $Y$, using complete intersections of two surfaces.
• Employment of the Schwarzenberger condition $c_2(c_2 + 2) \equiv 0 \pmod{12}$ to restrict possible $c_2$ values for stable bundles.
• Use of cohomological vanishing and base-point-freeness conditions on $\mathcal{I}_C(c_1)$ to constrain existence.
• Construction of examples via liaison of disjoint lines or conics to produce globally generated bundles with prescribed $c_1, c_2$.
• Analysis of normalized bundles and stability, particularly for $c_1 = 5$, to rule out non-split bundles in certain cases.

Experimental results

Research questions

• RQ1 For $n \geq 4$, are there any indecomposable globally generated rank two vector bundles with $c_1 \leq 5$?
• RQ2 Which pairs $(c_1, c_2)$ with $c_1 \leq 5$ and $c_1 = 5$ in $\mathbb{P}^3$ support an indecomposable globally generated rank two vector bundle?
• RQ3 Does a globally generated rank two vector bundle with $c_1 = 5$, $c_2 = 12$ exist on $\mathbb{P}^3$?
• RQ4 What are the geometric constraints on the zero-locus curve $C$ of a general section of such a bundle?
• RQ5 Can the existence of such bundles be determined via cohomological or liaison-theoretic criteria?

Key findings

• For $n \geq 4$, every globally generated rank two vector bundle with $c_1 \leq 5$ splits as a direct sum of two line bundles.
• In $\mathbb{P}^3$, indecomposable globally generated rank two bundles exist only for $c_1 \in \{2,4,5\}$ and specific $c_2$ values: $(2,2)$, $(4,5)$, $(4,6)$, $(4,7)$, $(4,8)$, $(5,8)$, $(5,10)$, and $(5,12)$.
• For $c_1 = 4$, $c_2 = 8$, the zero-locus curve $C$ may be irreducible or a disjoint union of two smooth conics.
• For $c_1 = 5$, $c_2 = 8$ or $10$, such bundles exist and are constructed via liaison of $s$ disjoint lines ($s=2,3$) to $s$ disjoint conics.
• The case $(c_1, c_2) = (5,12)$ remains unresolved: while a curve $X$ of degree 13 and genus 10 with $\omega_X(-1)$ base-point-free exists, the condition $h^0(\mathcal{I}_X(5)) \geq 3$ cannot be verified.
• The Horrocks-Mumford bundle is ruled out as a candidate for $c_1 = 5$, $c_2 = 6$ due to non-global generation, and $c_2 = 4$ is excluded by the Schwarzenberger condition.