[Paper Review] The Torelli problem for Logarithmic bundles of hypersurface arrangements in the projective space

Elena Angelini|arXiv (Cornell University)|Jun 5, 2015

Algebraic Geometry and Number Theory16 references3 citations

TL;DR

This paper investigates the Torelli problem for logarithmic bundles associated with arrangements of hypersurfaces in projective space, focusing on hyperplanes, conics, and higher-degree hypersurfaces. It establishes conditions under which the logarithmic bundle determines the arrangement up to isomorphism, proving that for line-conic arrangements in ℙ² with normal crossings, isomorphic logarithmic bundles do not uniquely determine the arrangement, and providing explicit constructions via cubic polynomials and polarization conditions.

ABSTRACT

Let $ \mathcal{D} = \{D_{1}, \ldots, D_{\ell}\} $ be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space $ \mathbb{P}^{n} $ and let $ Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ be the logarithmic bundle attached to it. Our aim is to study the injectivity of the correspondence $ \mathcal{D} \longrightarrow Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $. In order to do that, we first show that $ Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ admits a resolution of length $ 1 $ depending on the degrees and on the equations of $ D_{1}, \ldots, D_{\ell} $. Then, we prove a Torelli type theorem when $ \mathcal{D} $ has a sufficiently large number of components of the same degree $ d $, by recovering them as unstable smooth irreducible degree-$d$ hypersurfaces of $ Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $. The cases of one quadric and a pair of quadrics in $ \mathbb{P}^{n} $ are not Torelli; in particular, through a duality argument, we prove that the isomorphism class of the logarithmic bundle attached to a pair of quadrics is determined by the tangent hyperplanes to the pair. Finally, by describing the moduli spaces containing $ Ω^{1}_{\mathbb{P}^{2}}(log \mathcal{D}) $, we show that some line-conic arrangements are not of Torelli type.

Motivation & Objective

• To determine when the logarithmic bundle Ω¹( log 𝒟 ) uniquely determines the arrangement 𝒟 of hypersurfaces in projective space.
• To extend the Torelli problem beyond hyperplane arrangements to conics and higher-degree hypersurfaces.
• To analyze the case of line-conic arrangements in ℙ² and identify conditions under which isomorphic logarithmic bundles do not imply isomorphic arrangements.
• To construct explicit cubic polynomials whose partial derivatives match polarization conditions derived from logarithmic bundles.

Proposed method

• Uses Saito's general definition of logarithmic differential forms for non-normal-crossings arrangements.
• Applies the concept of jumping lines and stability to analyze the structure of logarithmic bundles.
• Reduces the problem to solving a system of linear equations in coefficients of conics and cubic polynomials.
• Employs integration techniques on partial differential equations to reconstruct a cubic polynomial g from given coefficient conditions.
• Uses matrix representations (9×9) to analyze the solution space of the system, showing a one-dimensional solution space.
• Relies on Hermite's theorem (1868) to interpret nets of conics as polar conics with respect to a cubic curve.

Experimental results

Research questions

• RQ1 Under what conditions does the isomorphism class of the logarithmic bundle Ω¹( log 𝒟 ) determine the arrangement 𝒟 in ℙⁿ?
• RQ2 Can two non-isomorphic line-conic arrangements in ℙ² have isomorphic logarithmic bundles?
• RQ3 What is the dimension of the solution space for the system of equations linking coefficients of conics and cubic polynomials in the logarithmic bundle context?
• RQ4 How do polarization conditions ∂ᵢg = xᵢ∂ᵢf relate to the structure of logarithmic bundles?
• RQ5 To what extent can the logarithmic bundle of a line-conic arrangement fail to be Torelli?

Key findings

• For line-conic arrangements in ℙ² with normal crossings, isomorphic logarithmic bundles do not imply isomorphic arrangements, as shown by explicit examples.
• The system of equations linking coefficients of conics and cubic polynomials has a solution space of dimension one, indicating non-uniqueness in reconstruction.
• A cubic polynomial g can be explicitly constructed via integration of partial differential conditions, yielding a solution that satisfies ∂₀g = x₀∂₀f, ∂₁g = x₁∂₁f, ∂₂g = x₂∂₂f.
• The construction confirms that two different line-conic arrangements can yield isomorphic logarithmic bundles, even when the arrangements are not isomorphic.
• The result aligns with Hermite's theorem, showing that a net of conics arises as the net of polar conics with respect to a cubic curve.
• In the case of diagonalized conics, the logarithmic bundle is isomorphic to that of a Fermat cubic, explaining the non-uniqueness in the Torelli problem for such arrangements.