[Paper Review] Boundedness of minimal partial du Val resolutions of canonical surface foliations

Yenan Chen|arXiv (Cornell University)|Dec 6, 2019

Algebraic Geometry and Number Theory7 references4 citations

TL;DR

This paper establishes the boundedness of minimal partial du Val resolutions for canonical surface foliations of general type, proving that such foliated surfaces with a fixed Hilbert function form a bounded family. The key result shows that canonical models and their resolutions are bounded, with effective generation of pluricanonical systems away from cusp singularities, and uniform bounds on dihedral and cusp singularities.

ABSTRACT

In this paper, we prove the boundedness of foliated surfaces $(X,\mathscr{F})$ which are minimal partial du Val resolutions of canonical models $(X_c,\mathscr{F}_c)$ of general type. For applications, we show the boundedness of non-cusp singularities on canonical models of foliated surfaces of general type and the effective generation on the complement of the cusp singularities.

Motivation & Objective

• To establish boundedness of minimal partial du Val resolutions for canonical surface foliations of general type.
• To improve upon prior results on birational boundedness by showing boundedness of both canonical models and their resolutions.
• To provide effective bounds on the index and embedding dimension of dihedral singularities on canonical models.
• To confirm [HL21, Conjecture 1] by showing effective generation of pluricanonical systems away from cusp singularities.
• To establish uniform bounds on the structure of cusp singularities in the context of foliated surfaces of general type.

Proposed method

• Use of Mumford’s intersection theory on normal complete surfaces to define and compute intersection numbers for Weil divisors.
• Application of the duality between foliations and saturated rank-one subsheaves of the tangent sheaf to define foliations on singular surfaces.
• Leveraging the minimal resolution of cusp and dihedral singularities to analyze the exceptional divisors and their intersection matrices.
• Employing Zariski decomposition and pseudo-effectivity arguments to study the bigness and ampleness of divisors on the resolution.
• Using the boundedness of embedding dimensions and group actions on singularities to derive uniform bounds on indices and orders.
• Applying results from [HL21], [Fuj12], and [Lan01] to establish very ampleness of twisted canonical bundles on the complement of exceptional loci.

Experimental results

Research questions

• RQ1 Are minimal partial du Val resolutions of canonical surface foliations of general type bounded for a fixed Hilbert function?
• RQ2 Can effective bounds be established on the index and embedding dimension of dihedral singularities on canonical models of foliated surfaces?
• RQ3 Does the pluricanonical system |mK_F| define a birational map that is an isomorphism outside cusp singularities for sufficiently divisible m?
• RQ4 Is there a uniform bound on the structure of cusp singularities in the context of foliated surfaces of general type?
• RQ5 Can the moduli theory of foliated surfaces of general type be advanced via boundedness of resolutions and canonical models?

Key findings

• For any fixed Hilbert function P(m) = χ(X_c, mK_F_c), the set of canonical models (X_c, F_c) and their minimal partial du Val resolutions (X, F) form a bounded family.
• There exist uniform constants C_1 and C_2 such that the index and embedding dimension of each dihedral singularity on a canonical model are bounded by C_1 and C_2, respectively.
• For any canonical model (X_c, F_c) with κ(K_F_c) = 2 and fixed Hilbert function P(m), there exists an integer m_P such that |mK_F| with m divisible by m_P defines a birational map that is an isomorphism on the complement of the cusp singularities.
• The intersection matrix and number of components in the minimal resolution of cusp singularities are uniformly bounded across the family.
• The index of the foliation at each cusp singularity is bounded, and the group action on the exceptional divisor has bounded order.
• The divisor K_Y + βK_G is very ample on the complement of the exceptional locus of the minimal resolution g: Y → X, ensuring effective generation of the pluricanonical system away from cusps.