[Paper Review] Moduli of Representations, Quiver Grassmannians, and Hilbert Schemes

Lutz Hille|arXiv (Cornell University)|May 22, 2015

Algebraic structures and combinatorial models15 references12 citations

TL;DR

This paper demonstrates that any projective scheme can be realized as both a moduli space of representations and a quiver Grassmannian over a finite-dimensional algebra, using Beilinson's equivalence and Hilbert scheme tautology. The key result is a constructive, explicit isomorphism between such schemes and quiver Grassmannians over the Kronecker quiver, with minimal quiver size and relations derived from defining equations.

ABSTRACT

It is a well established fact, that any projective algebraic variety is a moduli space of representations over some finite dimensional algebra. This algebra can be chosen in several ways. The counterpart in algebraic geometry is tautological: every variety is its own Hilber scheme of sheaves of length one. This holds even scheme theoretic. We use Beilinson's equivalence to get similar results for finite dimensional algebras, including moduli spaces and quiver grassmannians. Moreover, we show that several already known results can be traced back to the Hilbert scheme construction and Beilinson's equivalence.

Motivation & Objective

• To show that every projective scheme is isomorphic to a quiver Grassmannian of submodules of a specific module over a finite-dimensional algebra.
• To realize any projective scheme as a moduli space of indecomposable representations of dimension vector (1,…,1) over a Beilinson-type algebra with relations derived from defining equations.
• To minimize the quiver size, showing that the (n+1)-Kronecker quiver suffices for any projective subscheme of ℙⁿ.
• To unify and generalize known results on quiver Grassmannians and moduli spaces by tracing them back to the tautological Hilbert scheme construction and Beilinson’s equivalence.
• To extend the construction to work over arbitrary commutative rings, including ℤ, preserving integrality and functoriality.

Proposed method

• Use Beilinson’s equivalence to construct a finite-dimensional algebra A = kQ/J from a tilting bundle on ℙⁿ, extending it to include higher twists.
• Define the ideal J in the path algebra kQ as generated by the homogeneous relations fi, so that A = kQ/J parametrizes modules satisfying the defining equations of X.
• Construct a module M over A with dimension vector (1,…,1), such that the moduli space of its indecomposable representations is isomorphic to X.
• Show that the quiver Grassmannian of submodules of M of dimension vector (1,…,1) is isomorphic to X, using the fact that such submodules correspond to points in X.
• Reduce the quiver to the (n+1)-Kronecker quiver by choosing a suitable bundle T = ℌ(𝒪 ⊕ 𝒪(e) ⊕ 𝒪(d)) with d ≥ deg(fi), leading to a two-dimensional module over the Kronecker algebra.
• Prove that the Grassmannian of (1,1)-dimensional submodules over the Kronecker algebra is isomorphic to X by showing that each such submodule lifts uniquely to a (1,1,1)-submodule in the three-vertex quiver setting.

Experimental results

Research questions

• RQ1 Can every projective scheme be realized as a quiver Grassmannian of submodules of a finite-dimensional algebra?
• RQ2 Is it possible to construct such a realization using the minimal possible quiver, specifically the (n+1)-Kronecker quiver?
• RQ3 How can Beilinson’s equivalence and the Hilbert scheme tautology be combined to yield a uniform, constructive framework for such realizations?
• RQ4 Can the construction be extended to work over arbitrary commutative rings, including ℤ, preserving integrality and functoriality?
• RQ5 What is the minimal quiver size for realizing a given projective scheme as a quiver Grassmannian, and can further reduction beyond two vertices be achieved?

Key findings

• Any projective scheme X ⊂ ℙⁿ defined by homogeneous polynomials fi is isomorphic to the quiver Grassmannian of submodules of dimension vector (1,1) over the (n+1)-Kronecker algebra, with the module M = (S^{d-1}V/I_{d-1}, S^dV/I_d).
• The moduli space of all indecomposable representations of dimension vector (1,…,1) over the enlarged Beilinson algebra kQ/J is isomorphic to X as a scheme.
• The construction is explicit and functorial: the quiver Q is the Beilinson quiver, J is generated by the fi, and M is the unique sincere injective cover of a simple module.
• The isomorphism between the Grassmannian of (1,1,1)-submodules in a three-vertex quiver and the Grassmannian of (1,1)-submodules in the Kronecker quiver is a bijection due to the one-dimensionality of the top component.
• The construction works over any base field and even over ℤ, with all objects defined integrally via symmetric powers S^mV and quotients by ideals I_m.
• The result generalizes and unifies previous constructions: it recovers known results on quiver Grassmannians and moduli spaces as special cases of the Hilbert scheme tautology and Beilinson’s equivalence.