[Paper Review] Moore Matrices and Ulrich Bundles on an Elliptic Curve

Ragnar-Olaf Buchweitz, Alexander Pavlov|arXiv (Cornell University)|Nov 17, 2015

Algebraic structures and combinatorial models13 references3 citations

TL;DR

This paper provides normal forms for linear determinantal representations of smooth elliptic curves in Hesse form using Moore matrices, and constructs matrix factorizations for all indecomposable rank-2, degree-0 Ulrich bundles (without nonzero sections) on such curves. It establishes that these bundles correspond to extensions of line bundles of degree 0 with trivial global sections, realized via Moore matrix constructions and divergence maps.

ABSTRACT

We give normal forms of determinantal representations of a smooth projective plane cubic in terms of Moore matrices. Building on this, we exhibit matrix factorizations for all indecomposable vector bundles of rank 2 and degree 0 without nonzero sections, also called Ulrich bundles, on such curves.

Motivation & Objective

• To provide explicit normal forms for linear determinantal representations of smooth plane cubic curves in Hesse form over fields of characteristic ≠ 2,3.
• To classify all indecomposable rank-2, degree-0 vector bundles without nonzero global sections (Ulrich bundles) on such curves.
• To realize these Ulrich bundles via matrix factorizations constructed from Moore matrices and cokernel constructions.
• To establish a correspondence between such bundles and points on the elliptic curve via the group law, particularly using 3-torsion conditions.
• To compute the extension group of the associated line bundles using divergence maps on matrix spaces, showing it is one-dimensional over the base field.

Proposed method

• Utilize Moore matrices of the form $ M_{(a_0,a_1,a_2),old{x}} = (a_{i+j}x_{i-j}) $ indexed by points $[a_0:a_1:a_2] otin E$ with $a_0a_1a_2 \neq 0$.
• Show that every linear determinantal representation of the Hesse cubic is equivalent to such a Moore matrix under the action of $ \mathsf{G}_3 = \mathrm{GL}(3,K) \times \mathrm{GL}(3,K)/\mathbb{G}_m(K) $.
• Prove that two Moore matrices yield equivalent representations if and only if $ 3 \cdot_E \mathbf{a} = 3 \cdot_E \mathbf{a}' $, where $ \cdot_E $ denotes the elliptic curve group law with identity at $[0:-1:1]$.
• Construct matrix factorizations for Ulrich bundles by realizing them as cokernels of triangular block matrices built from Moore matrices and their adjugates.
• Realize the extension group $ \operatorname{Ext}^1_R(\mathbf{L},\mathbf{L}) $ as a quotient of matrix spaces modulo commutators, with the divergence map $ \mathsf{div} $ inducing an isomorphism to $ K $.
• Use Atiyah’s classification of rank-2 vector bundles on elliptic curves to show that all such Ulrich bundles arise as non-split extensions of line bundles of degree 0 with no global sections.

Experimental results

Research questions

• RQ1 What are the normal forms for linear determinantal representations of smooth elliptic curves in Hesse form?
• RQ2 How can all indecomposable rank-2, degree-0 Ulrich bundles on such curves be explicitly constructed via matrix factorizations?
• RQ3 What is the geometric and algebraic condition under which two Moore matrices yield equivalent determinantal representations?
• RQ4 How is the extension group of the associated line bundle realized in terms of matrix spaces and divergence maps?
• RQ5 What is the role of the 3-torsion condition $ 3 \cdot_E \mathbf{a} = 3 \cdot_E \mathbf{a}' $ in classifying these representations?

Key findings

• Every linear determinantal representation of a smooth Hesse cubic curve is equivalent to a Moore matrix $ M_{(a_0,a_1,a_2),old{x}} $ with $[a_0:a_1:a_2] \in E$ and $a_0a_1a_2 \neq 0$.
• Two such Moore matrices yield equivalent representations if and only if $ 3 \cdot_E \mathbf{a} = 3 \cdot_E \mathbf{a}' $, where $ \cdot_E $ is the group law on the elliptic curve with identity at $[0:-1:1]$.
• All indecomposable rank-2, degree-0 Ulrich bundles on the curve arise as non-split extensions of line bundles $ \mathbf{L} = \operatorname{Coker} M_{a,x} $ with $ \deg \mathbf{L} = 0 $ and $ H^0(E, \mathbf{L}) = 0 $.
• The extension group $ \operatorname{Ext}^1_R(\mathbf{L},\mathbf{L}) $ is one-dimensional over $ K $, and is isomorphic to $ K $ via the divergence map $ \mathsf{div} $ on matrices $ M_{b,y} $.
• The matrix factorization for such Ulrich bundles is given by a triangular block matrix whose cokernel is the desired bundle, with the non-splitness ensured by $ \mathsf{div}(M_{a,x}) = 3 \neq 0 $.
• The construction realizes all such Ulrich bundles via a canonical correspondence with points $ \mathbf{a} \in E $ satisfying $ a_0a_1a_2 \neq 0 $, excluding the 3-torsion points where the product vanishes.