[Paper Review] On the moduli space of semi-stable plane sheaves with Euler characteristic one and supported on sextic curves

Mario Maican|arXiv (Cornell University)|Apr 30, 2011

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper studies the moduli space of Gieseker semi-stable sheaves on the complex projective plane supported on sextic curves with Euler characteristic one. It provides locally free resolutions of length one for all such sheaves, decomposes the moduli space into five strata via cohomological conditions, and describes each stratum as a geometric quotient of a parameter space by an algebraic group action, offering explicit geometric and cohomological characterizations of the sheaves in each stratum.

ABSTRACT

We study the moduli space of Gieseker semi-stable sheaves on the complex projective plane supported on sextic curves and having Euler characteristic one. We determine locally free resolutions of length one for all such sheaves. We decompose the moduli space into strata which occur naturally as quotients modulo actions of certain algebraic groups. In some cases we give concrete geometric descriptions of the strata.

Motivation & Objective

• To understand the geometry of the moduli space M_{P²}(6,1) parametrizing Gieseker semi-stable sheaves on P² with Hilbert polynomial P(m) = 6m + 1.
• To decompose this moduli space into strata based on cohomological invariants, each corresponding to distinct sheaf types.
• To construct locally free resolutions of length one for all such sheaves, enabling explicit geometric descriptions of the strata.
• To describe each stratum as a geometric quotient of a parameter space of morphisms by an algebraic group action.
• To relate the strata to dual moduli spaces via the duality isomorphism M_{P²}(6,1) ≅ M_{P²}(6,5), extending known results for lower-degree curves.

Proposed method

• The moduli space M_{P²}(6,1) is stratified into five locally closed subsets X₀ through X₅ based on cohomological conditions on the sheaves.
• Each stratum Xi is realized as a geometric quotient of a locally closed subset Wi ⊂ Hom(Ai, Bi) by the action of Aut(Ai) × Aut(Bi), where Ai and Bi are direct sums of line bundles.
• Locally free resolutions of length one are constructed for all sheaves in each stratum, with the sheaf being the cokernel of a morphism φ ∈ Hom(Ai, Bi).
• The strata are described via fiber bundle structures: X₀ over N(3,5,4), X₂ over Y × P², X₃ over P² × N(3,2,3), X₄ birational to a fiber bundle over Grass(2,6), and X₅ isomorphic to a Hilbert flag scheme.
• Duality is used to relate M_{P²}(6,1) to M_{P²}(6,5), with dual strata X_i^D defined by dual cohomological conditions.
• Cohomological vanishing and stability arguments are used to rule out destabilizing subsheaves, ensuring the strata are well-defined and irreducible.

Experimental results

Research questions

• RQ1 What is the geometric structure of the moduli space M_{P²}(6,1) of semi-stable sheaves supported on sextic curves with Euler characteristic one?
• RQ2 How can the moduli space be stratified according to cohomological invariants of the sheaves it parametrizes?
• RQ3 What are the explicit locally free resolutions of length one for all such sheaves, and how do they realize the strata as quotients of morphism spaces?
• RQ4 How do the strata relate to known moduli spaces such as N(3,5,4), N(3,2,3), and Grass(2,6)?
• RQ5 What is the role of duality in relating M_{P²}(6,1) to M_{P²}(6,5), and how are the dual strata characterized?

Key findings

• The moduli space M_{P²}(6,1) is a smooth, irreducible, rational projective variety of dimension 37.
• The space is stratified into five strata: X₀ (open, codimension 0), X₁ and X₂ (codimension 2), X₃ ∪ X₄ (codimension 6), and X₅ (codimension 8).
• X₀ is an open subset of a fiber bundle with fiber P¹⁷ over the moduli space N(3,5,4) of semi-stable Kronecker modules f: 5O(-2) → 4O(-1).
• X₅ is isomorphic to the Hilbert flag scheme parametrizing sextic curves with a zero-dimensional subscheme of length 2.
• The generic sheaves in X₃ are of the form O_C(2)(-P₁ - P₂ - P₃ + P₄), where P₁,P₂,P₃ are non-collinear points on a smooth sextic C.
• The generic sheaves in X₄ are of the form O_C(1)(P₁ + P₂ + P₃ + P₄), where no three of the four points are collinear on C.