[Paper Review] Globally Generated Vector Bundles on P^n with c_1=3

Cristian Anghel, Iustin Coandă+1 more|arXiv (Cornell University)|Feb 28, 2012

Black Holes and Theoretical Physics31 references19 citations

TL;DR

This paper classifies globally generated vector bundles of first Chern class $c_1 = 3$ on projective spaces $\mathbb{P}^n$ for $n \neq 3$, using Serre's correspondence and its higher-rank generalizations. It provides complete resolutions for such bundles on $\mathbb{P}^2$, $\mathbb{P}^4$, $\mathbb{P}^5$, and establishes a recursive structure for $n \geq 5$, showing all higher-rank bundles arise as extensions of lower-rank ones by trivial bundles.

ABSTRACT

One classifies the globally generated vector bundles on P^n (n ot = 3) with the first Chern class c_1 = 3. The case n = 3 is treated in arXiv:1202.5988 [math.AG]. The case c_1 = 2 was treated by J.C. Sierra and L. Ugaglia (see References), the case c_1 = 3, rank = 2 is settled by S. Huh (see References), the case rank = 2, c_1 \le 5 is studied by L. Chiodera and Ph. Ellia (see References).

Motivation & Objective

• To classify globally generated vector bundles of first Chern class $c_1 = 3$ on $\mathbb{P}^n$ for $n \neq 3$, extending prior classifications for $c_1 = 1, 2$ and rank 2 bundles.
• To understand the structure of such bundles using Serre's correspondence and its higher-rank generalization via Vogelaar's work.
• To determine the complete list of such bundles of rank at most $n$ on $\mathbb{P}^n$ for $n = 2, 4, 5$, and to describe the structure of higher-rank bundles.
• To establish that for $n \geq 5$, all globally generated bundles with $c_1 = 3$ and rank $r > n$ are extensions of rank-$n$ bundles by trivial bundles.

Proposed method

• Utilizes Serre's theorem relating rank-2 vector bundles to codimension-2 locally complete intersection subschemes, generalized to higher ranks via Vogelaar's framework.
• Employs cohomological vanishing arguments and the splitting criterion of Horrocks to prove that bundles with certain vanishing cohomology admit resolutions by direct sums of line bundles.
• Applies the Beilinson spectral sequence to analyze the structure of bundles like $\Omega^1(2)$ on $\mathbb{P}^4$, showing their degeneracy and resolution properties.
• Analyzes the degree and geometric type of the associated subschemes $Z$ (e.g., elliptic scrolls, Bordiga varieties, complete intersections) to classify possible resolutions.
• Uses induction and restriction to hyperplanes to extend classification from $\mathbb{P}^n$ to $\mathbb{P}^{n+1}$, proving the recursive structure for $n \geq 5$.
• Relies on known resolutions of ideal sheaves $I_Z$ for subschemes $Z$ of degree $d = 5, 6, 7, 9$, and constructs corresponding vector bundles via cokernel sequences.

Experimental results

Research questions

• RQ1 What are the globally generated vector bundles of $c_1 = 3$ on $\mathbb{P}^2$ that are not direct sums of line bundles?
• RQ2 How do the globally generated vector bundles of $c_1 = 3$ on $\mathbb{P}^4$ and $\mathbb{P}^5$ with rank $\leq n$ arise from geometric subschemes?
• RQ3 What is the structure of globally generated vector bundles with $c_1 = 3$ on $\mathbb{P}^n$ for $n \geq 5$, particularly for ranks exceeding $n$?
• RQ4 Can the classification of such bundles be extended inductively from $\mathbb{P}^n$ to $\mathbb{P}^{n+1}$ using cohomological restriction theorems?
• RQ5 How do the resolutions of these bundles relate to the geometry of the associated subschemes $Z$ of codimension 2?

Key findings

• On $\mathbb{P}^2$, there are exactly eight non-split, globally generated rank-2 vector bundles with $c_1 = 3$, all given by explicit monad-type resolutions involving line bundles and twists.
• On $\mathbb{P}^4$, there are no globally generated rank-2 bundles with $c_1 = 3$, but there exists a unique rank-3 bundle (dual of the Trautmann-Vetter bundle) and several rank-4 bundles, all arising from resolutions with specific kernel structures.
• On $\mathbb{P}^5$, there are no globally generated bundles of rank 2, 3, or 4 with $c_1 = 3$, but there are seven distinct rank-5 bundles, all given by monad-type resolutions with specific kernel types.
• For $n \geq 5$, all globally generated vector bundles with $c_1 = 3$ and rank $r > n$ are extensions of rank-$n$ bundles by trivial bundles, and are classified by seven standard resolution types.
• The classification for $n \geq 5$ is complete and recursive: all such bundles arise from the seven resolution types involving $\mathcal{O}(-1)$, $\mathcal{O}(-2)$, $\mathcal{O}(-3)$, and $\mathcal{O}(-1)\oplus\mathcal{O}(-2)$ kernels with appropriate trivial summands.
• The bundle $\Omega^1(2)$ on $\mathbb{P}^4$ is isomorphic to the dual of the Trautmann-Vetter bundle and arises from a degenerate Beilinson spectral sequence, confirming its resolution and global generation.