[Paper Review] Some (big) irreducible components of the moduli space of minimal surfaces of general type with $p_g=q=1$ and $K^2=4$

Roberto Pignatelli|ArXiv.org|Jan 8, 2008

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper constructs eight unirational families of minimal surfaces of general type with $p_g = q = 1$ and $K^2 = 4$, all of which are irreducible components of the moduli space. The authors prove these components have dimension 5 or 4—strictly larger than the expected dimension—under the condition that the direct image of the bicanonical sheaf under the Albanese fibration is a direct sum of line bundles.

ABSTRACT

In this paper we study the minimal surfaces of general type with $p_g=q=1$ and $K^2=4$ whose Albanese general fibre has genus 2, classifying those such that the direct image (under the Albanese morphism) of the bicanonical sheaf is sum of line bundles. We find 8 unirational families, all of dimension strictly bigger than the expected one. These families are pairwise disjoint irreducible components of the moduli space of minimal surfaces of general type.

Motivation & Objective

• To classify minimal surfaces of general type with $p_g = q = 1$ and $K^2 = 4$ whose Albanese fibration has genus 2 fibers.
• To study the structure of the direct image $\alpha_*\omega_S^2$ under the Albanese morphism $\alpha$.
• To determine whether the condition that $\alpha_*\omega_S^2$ is a direct sum of line bundles yields irreducible components of the moduli space.
• To explain why the dimension of these components exceeds the expected dimension from deformation theory.
• To compare the constructed families with existing constructions, particularly Polizzi's 4-dimensional family.

Proposed method

• Applies the structure theorem for genus 2 fibrations to analyze the relative canonical algebra of the Albanese fibration.
• Uses the relative bicanonical map and the geometry of a conic bundle obtained as the quotient by the hyperelliptic involution.
• Analyzes the intersection of bicanonical curves with the critical locus of the Albanese morphism to bound $h^1(\mathcal{T}_S)$.
• Employs the condition that $\alpha_*\omega_S^2$ is a direct sum of line bundles as a key technical assumption to classify possible families.
• Compares the constructed families with Polizzi’s construction by analyzing the presence of nodes and the structure of the conic bundle.
• Uses cohomological bounds on $h^1(\mathcal{T}_S)$ to prove irreducibility of the components, relating it to subsystems of the bicanonical system.

Experimental results

Research questions

• RQ1 Are the families of surfaces with $p_g = q = 1$, $K^2 = 4$, and $\alpha_*\omega_S^2$ decomposable into line bundles irreducible components of the moduli space?
• RQ2 Why do all eight constructed families have dimension strictly greater than the expected dimension $3$?
• RQ3 Is the number of direct summands of $\alpha_*\omega_S^2$ a deformation or topological invariant?
• RQ4 How does Polizzi’s 4-dimensional family of nodal surfaces relate to the new families constructed here?
• RQ5 Can the condition that $\alpha_*\omega_S^2$ is a direct sum of line bundles be relaxed while preserving irreducibility of components?

Key findings

• The moduli space contains eight irreducible components: one of dimension 5 and seven of dimension 4.
• All eight families are unirational, and the general surface in each has ample canonical class.
• The dimension of each component exceeds the expected lower bound of 3, which is derived from standard deformation theory.
• Polizzi’s 4-dimensional family of nodal surfaces is a proper subfamily of the 5-dimensional component $\mathcal{M}_{2,3}$, with codimension 1.
• The condition that $\alpha_*\omega_S^2$ is a direct sum of line bundles, though closed, leads to irreducible components, which is unexpected.
• The bound $h^1(\mathcal{T}_S) \leq 5$ is achieved only in the $\mathcal{M}_{2,3}$ family, where a pencil of bicanonical curves through the critical locus exists.