[Paper Review] Resolutions of homogeneous bundles on P^2

Giorgio Ottaviani, Elena Rubei|ArXiv.org|Jan 28, 2004

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper provides a complete classification of homogeneous vector bundles on $\mathbb{P}^2$ via their minimal free resolutions, characterizing the $SL(V)$-representations and $SL(V)$-invariant maps that can appear in such resolutions. The key contribution is a criterion: a minimal free resolution $0 \to A \to B \to E \to 0$ exists for a homogeneous bundle $E$ if and only if the induced map $M(\alpha)$ on representation coefficients is injective and zero on diagonal components, enabling a parametrization of bundles via injective matrices with prescribed symmetry.

ABSTRACT

In this paper we characterize minimal free resolutions of homogeneous bundles on P^2. Besides we study stability and simplicity of homogeneous bundles on P^2 by means of their minimal free resolutions; in particular we give a criterion to see when a homogeneous bundle is simple by means of its minimal free resolution in the case the first bundle of the resolution is irreducible.

Motivation & Objective

• To classify homogeneous vector bundles on $\mathbb{P}^2$ by their minimal free resolutions, avoiding direct classification via parabolic subgroup representations.
• To characterize the $SL(V)$-representations $A_q$, $B_q$ and $SL(V)$-invariant maps that can occur in such resolutions.
• To provide a parametrization of homogeneous bundles using injective matrices of representation coefficients, up to invertible transformations with specific symmetry.
• To study simplicity and stability of regular elementary homogeneous bundles using quiver representations and Rohmfeld's criterion.

Proposed method

• Use of minimal free resolutions of the form $0 \to \bigoplus_q \mathcal{O}(-q) \otimes_{\mathbb{C}} A_q \to \bigoplus_q \mathcal{O}(-q) \otimes_{\mathbb{C}} B_q \to E \to 0$, where $A_q, B_q$ are $SL(V)$-representations.
• Definition of the map $M(\gamma)$ that extracts the coefficient map from an $SL(V)$-invariant map $\gamma$ between tensor products of Schur functors and line bundles.
• Application of Pieri’s formula and Schur functor properties to analyze existence and vanishing of $SL(V)$-invariant maps between symmetric powers of $V$.
• Use of quiver representations and subrepresentation techniques to analyze subbundles and apply Rohmfeld’s criterion for semistability.
• Construction of explicit examples via maps $\varphi_{\rho,p}$, which are unique up to scalar and satisfy a composition property $\varphi_{\rho,p} = \varphi_{\rho,p'} \circ \varphi_{p',p}$ for $\rho \geq p' \geq p$.
• Proof of injectivity of $M(\alpha)$ as a necessary and sufficient condition for the injectivity of the resolution map $\alpha$ via fiber-wise analysis and linear dependence arguments.

Experimental results

Research questions

• RQ1 Which $SL(V)$-representations and $SL(V)$-invariant maps can appear in the minimal free resolutions of homogeneous vector bundles on $\mathbb{P}^2$?
• RQ2 What conditions on the coefficient maps $M(\alpha)$ ensure that a given resolution $0 \to A \to B \to E \to 0$ defines a homogeneous bundle?
• RQ3 When is a regular elementary homogeneous bundle on $\mathbb{P}^2$ simple or stable?
• RQ4 How can quiver representations be used to characterize subbundles and test stability of homogeneous bundles?
• RQ5 Can the set of all homogeneous bundles on $\mathbb{P}^2$ be parametrized by injective matrices of representation coefficients with prescribed symmetry?

Key findings

• A minimal free resolution $0 \to A \to B \to E \to 0$ of a homogeneous bundle $E$ on $\mathbb{P}^2$ exists if and only if $\dim(\bigoplus_{p \geq \tilde{p}} A_{c+p}^{p,q}) \leq \dim(\bigoplus_{\rho > \tilde{p}} B_{c+\rho}^{\rho,q})$ for all $c \in \mathbb{Z}$, $q, \tilde{p} \in \mathbb{N}$.
• An $SL(V)$-invariant map $\alpha: A \to B$ induces a minimal free resolution of a homogeneous bundle $E$ if and only if $M(\alpha_{p,q,r}) = 0$ for all $p,q,r$ and $M(\alpha_{q,r})$ is injective for all $q,r$.
• The simplest regular elementary homogeneous bundles defined by $0 \to S^{p,q}V \otimes \mathcal{O}(-s) \xrightarrow{\varphi} S^{p+s,q}V \otimes \mathcal{O} \to E \to 0$ are stable for $p \geq q$, $s \in \mathbb{N}$, with $\varphi$ nonzero and $SL(V)$-invariant.
• A regular elementary homogeneous bundle $E$ is simple if and only if its resolution is of the form $0 \to S^{p,q}V \otimes \mathcal{O}(-s) \xrightarrow{\varphi} W \otimes \mathcal{O} \to E \to 0$, where $W \subset S^{p,q}V \otimes S^sV$ is an $SL(V)$-submodule, all components of $\varphi$ are nonzero $SL(V)$-invariant maps, and the resolution satisfies the $M(\alpha)$-injectivity condition.
• The map $M(\alpha)$ is injective if and only if the original map $\alpha$ is injective, as shown via fiber-wise analysis and linear dependence arguments on the coefficients of the resolution.
• The existence of a nonzero element $y$ in $S^{\lambda_1,\dots,\lambda_n}V$ such that $\varphi(y) = 0$ for all $SL(V)$-invariant maps $\varphi$ of the form $S^{\lambda_1,\dots,\lambda_n}V(t) \to S^{\lambda_1+s_1,\dots,\lambda_n+s_n}V(t + s_1 + \cdots + s_n)$ with $s_2 + \cdots + s_n > 0$ is proven using Pieri’s formula and duality.