[Paper Review] Geomety of generic Moishezon twistor spaces on 4CP^2

Nobuhiro Honda|arXiv (Cornell University)|Sep 16, 2010

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper constructs generic Moishezon twistor spaces on $4\mathbb{CP}^2$ as double covers over a scroll of 2-planes in $\mathbb{CP}^4$ over a conic, explicitly determining the branch divisor's defining equation as $z_0z_3z_4f = Q^2$, where $f$ is linear and $Q$ quadratic with real coefficients. The key contribution is the explicit description of the moduli space dimension and the identification of singularities on the branch divisor, including 13 ordinary double points and two $A_3$-singularities.

ABSTRACT

In this paper we investigate a family of Moishezon twistor spaces on the connected sum of 4 complex projective planes, which can be regarded as a direct generalization of the twistor spaces on 3CP^2 of double solid type studied by Poon and Kreussler-Kurke. These twistor spaces have a natural structure of double covering over a scroll of 2-planes over a conic. We determine the defining equations of the branch divisors in an explicit form, which are very similar to the case of 3CP^2. Using these explicit description we compute the dimension of the moduli spaces of these twistor spaces. Also we observe that similarly to the case of 3CP^2, these twistor spaces can also be considered as generic Moishezon twistor spaces on 4CP^2. We obtain these results by analyzing the anticanonical map of the twistor spaces in detail, which enables us to give an explicit construction of the twistor spaces, up to small resolutions.

Motivation & Objective

• To construct generic Moishezon twistor spaces on $4\mathbb{CP}^2$ as a natural generalization of those on $3\mathbb{CP}^2$ studied by Poon and Kreussler-Kurke.
• To determine the defining equations of the branch divisors of the anticanonical double covering structure on these twistor spaces.
• To compute the dimension of the moduli space of such twistor spaces using detailed analysis of the anticanonical system.
• To classify the singularities of the branch divisor and identify their invariants, including ordinary double points and $A_3$-singularities.

Proposed method

• Analyzing the anticanonical system of the twistor space to construct a rational map to $\mathbb{CP}^4$ whose image is a scroll $Y$ of 2-planes over a conic.
• Explicitly resolving the indeterminacy locus of the anticanonical map via blowups and blowdowns to obtain a degree-2 morphism $Z_1 \to Y$.
• Identifying the branch divisor $B$ of the double covering as the intersection of $Y$ with a quartic hypersurface in $\mathbb{CP}^4$.
• Finding five hyperplanes in $\mathbb{CP}^4$ such that $B$ intersects them in double curves, analogous to conics in the $3\mathbb{CP}^2$ case.
• Proving that the five double curves lie on a unique quadratic hypersurface, leading to the key equation $z_0z_3z_4f = Q^2$ for the quartic hypersurface.
• Using coordinate transformations and local blowups to invert the contraction map $Z_3 \to Z_4$, modeling a singular version of Hironaka’s non-projective 3-fold construction.

Experimental results

Research questions

• RQ1 What is the explicit defining equation of the branch divisor for generic Moishezon twistor spaces on $4\mathbb{CP}^2$?
• RQ2 How does the anticanonical map of these twistor spaces factor, and what is the structure of the resulting double covering?
• RQ3 What are the singularities of the branch divisor, and how do they compare to those in the $3\mathbb{CP}^2$ case?
• RQ4 What is the dimension of the moduli space of such twistor spaces?
• RQ5 Can the double covering structure be realized via a small resolution of singularities, and how does this relate to known constructions in complex geometry?

Key findings

• The branch divisor $B$ of the double covering is defined by the equation $z_0z_3z_4f = Q^2$ in $\mathbb{CP}^4$, where $f$ is linear and $Q$ is a quadratic polynomial with real coefficients.
• The branch divisor $B$ has exactly 13 ordinary double points and two $A_3$-singularities, with additional isolated singularities whose invariants are fully determined.
• The moduli space of these generic Moishezon twistor spaces on $4\mathbb{CP}^2$ has dimension 10, computed via the anticanonical system analysis.
• The double covering structure arises from a degree-2 morphism $Z_1 \to Y$, where $Y$ is a scroll of 2-planes over a conic in $\mathbb{CP}^4$, and the indeterminacy locus is resolved via explicit blowups.
• The singularities of the branch divisor are analyzed in detail: 5 double curves exist, and their intersections yield the 13 ordinary double points and two $A_3$-points.
• The construction of the twistor space via small resolution and blowup at a reducible curve provides a singular analog of Hironaka’s non-projective 3-fold, confirming the birational geometry of the model.