[Paper Review] Reider's Theorem and Thaddeus Pairs Revisited

Daniele Arcara, Aaron Bertram|ArXiv.org|Apr 22, 2009

Algebraic Geometry and Number Theory7 references16 citations

TL;DR

This paper reinterprets Reider's theorem and Thaddeus pairs using Bridgeland stability conditions in higher dimensions, generalizing the classical results on surfaces to threefolds. It establishes a framework where vanishing of Ext groups is implied by Bridgeland slope inequalities, leading to a higher-dimensional Reider-type theorem and a new moduli-theoretic interpretation of Thaddeus pairs via stable objects in tilted categories.

ABSTRACT

Bridgeland stability conditions allow for a new generalization of Thaddeus pairs to surfaces and a new interpretation of Reider's theorem as a consequence of "Schur's lemma" for stable objects (Hom(E,F) = 0 if E,F are stable objects and the slope of E exceeds the slope of F). One improvement of Reider's theorem results (Proposition 3.8/Corollary 3.9), and wall-crossings for the new Thaddeus pairs are discussed. This paper was submitted to the CMI conference proceedings celebrating the 65th birthday of Peter Newstead.

Motivation & Objective

• To generalize Thaddeus pairs and Reider's theorem from curves and surfaces to higher-dimensional varieties using Bridgeland stability conditions.
• To establish a higher-dimensional analog of Reider's theorem by linking vanishing of Ext^1 groups to Bridgeland slope inequalities.
• To investigate the moduli of Bridgeland-stable objects and their birational geometry in the context of Mukai flops and wall-crossing.
• To explore whether Bridgeland-stable objects of fixed numerical class form projective or proper algebraic spaces.
• To investigate the possibility of a codimension-three Bogomolov-type inequality for threefolds, potentially leading to a proof of Fujita's conjecture.

Proposed method

• Reinterprets Reider's argument via Bridgeland stability in tilted categories A_s within the derived category D(X), replacing Mumford stability with Bridgeland stability.
• Uses the Bogomolov inequality and Hodge Index Theorem to construct Bridgeland stability conditions on smooth projective surfaces.
• Analyzes the stability of objects L⊗I_Z and I_W^∨[1] in A_s, with vanishing of Hom(L⊗I_Z, I_W^∨[1][1]) implied by slope inequality μ(L⊗I_Z) > μ(I_W^∨[1]).
• Generalizes Thaddeus pairs to surfaces as extensions in A_s: 0 → I_W^∨[1] → E_ε^• → L⊗I_Z → 0, with stability requiring slope inequality.
• Applies the framework to threefolds with Pic(X) = Z·H, studying extensions 0 → O_X[1] → E_ε^• → L⊗I_Z → 0 in A_{1/2}.
• Proposes that instability of E_ε^• for large d and small t may be governed by a codimension-three Bogomolov-type inequality.

Experimental results

Research questions

• RQ1 Can stable Thaddeus pairs be defined as a moduli problem for varying t, and is this moduli space projective or smooth?
• RQ2 Are isomorphism classes of Bridgeland-stable objects of fixed numerical type represented by a (quasi)-projective scheme of finite type?
• RQ3 Is there an example where such isomorphism classes form a proper algebraic space that is not a projective scheme?
• RQ4 Do bounds d₀ and t₀ exist such that all E_ε^• are μ_{1/2+it}-unstable when d > d₀ and t < t₀ in threefolds with Pic(X) = Z·H?
• RQ5 Can the 'interesting cases' of destabilizing subobjects in threefolds (e.g., rank r torsion-free sheaves with c₁ = (r+1)/2 H) be numerically eliminated via Hodge Index-type arguments?

Key findings

• The vanishing of Ext¹(L⊗I_Z, O_X) is implied by the Bridgeland slope inequality μ(L⊗I_Z) > μ(O_X[1]), a stronger condition than vanishing alone.
• For surfaces, the space of t-stable Thaddeus pairs is defined as the proper transform of P(Ext¹(L,O_S[1])) in the moduli space of μ_{1/2+it}-stable objects with invariants (0,H,H²/2).
• When K_S = 0, the moduli space undergoes Mukai flops via wall-crossing, replacing projective bundles over Hilb^d(S)×Hilb^d(S) with dual bundles.
• The derived dual I_W^∨[1] generalizes the line bundle O_C(W) in the curve case, enabling a higher-dimensional Thaddeus pair construction.
• In threefolds with Pic(X) = Z·H, the existence of destabilizing subobjects K ⊂ L⊗I_Z in A_{1/2} is constrained by numerical conditions, including rank r odd and c₁(K) = (r+1)/2 H.
• The paper suggests that a codimension-three Bogomolov-type inequality could imply Fujita’s conjecture for threefolds, should such an inequality be established.