[Paper Review] Extending hyperelliptic K3 surfaces, and Godeaux surfaces with torsion Z/2

Stephen Coughlan|ArXiv.org|Sep 30, 2009

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper constructs an 8-parameter family of Godeaux surfaces with ℤ/2 torsion by extending hyperelliptic K3 surfaces to a Fano 6-fold via projection-unprojection techniques, using the Fano 6-fold as a 'key variety' to realize hyperelliptic surfaces of general type with p_g=1, K²=2, and then taking ℤ/2 quotients. The method establishes a one-to-one correspondence between such K3 surfaces and their Fano 6-fold extensions.

ABSTRACT

We study the extension of a hyperelliptic K3 surface to a Fano 6-fold. This determines a family of surfaces of general type with p_g=1, K^2=2 and hyperelliptic canonical curve, where each surface is a weighted complete intersection inside a Fano 6-fold. Finally, we use these hyperelliptic surfaces to determine an 8-parameter family of Godeaux surfaces with torsion Z/2.

Motivation & Objective

• To extend hyperelliptic K3 surfaces to Fano 6-folds using a projection-unprojection construction.
• To construct a 15-parameter family of hyperelliptic surfaces of general type with p_g=1, K²=2, and trivial torsion.
• To use these surfaces as a foundation for constructing Godeaux surfaces with ℤ/2 torsion via ℤ/2 quotients.
• To establish a geometric correspondence between the moduli of hyperelliptic K3 surfaces and their Fano 6-fold extensions.
• To provide a new construction of Godeaux surfaces with ℤ/2 torsion that may yield an irreducible 8-dimensional component in the moduli space.

Proposed method

• Use projection-unprojection techniques to extend a hyperelliptic K3 surface T ⊂ ℙ(2⁴,3⁴,4) to a Fano 6-fold W ⊂ ℙ(1⁴,2⁴,3⁴,4) with 10×1/2 points.
• Construct a tower of varieties D ⊂ T ⊂ W³ ⊂ W⁴ ⊂ W⁵ ⊂ W⁶, where each W^i is a weighted complete intersection in the next.
• Lift the hyperelliptic involution on the genus 3 curve D to the K3 surface T and then to the Fano 6-fold W via equivariant unprojection.
• Define the Godeaux involution on the Fano 6-fold by ensuring the involution acts on the ring R(W,A) with a ℤ⊕ℤ/2-bigrading.
• Construct the Godeaux surface X as a ℤ/2 quotient of a hyperelliptic surface Y, where Y is a complete intersection of type (1⁺,1⁺,1⁻,2⁻) in W.
• Ensure the complete intersection Y avoids the 10×1/2 singular points of W, which is an open condition guaranteeing smoothness of the quotient.

Experimental results

Research questions

• RQ1 Can hyperelliptic K3 surfaces with 10×1/2 singularities be uniquely extended to quasismooth Fano 6-folds?
• RQ2 What is the moduli dimension of the family of hyperelliptic surfaces of general type with p_g=1, K²=2, and trivial torsion?
• RQ3 Can the Godeaux involution on a hyperelliptic curve be lifted to a Fano 6-fold and extended to a fixed-point-free involution on a complete intersection?
• RQ4 Does the construction yield an 8-dimensional irreducible component in the moduli space of Godeaux surfaces with ℤ/2 torsion?
• RQ5 How does the key variety method via Fano 6-folds facilitate the construction of surfaces of general type with specified invariants and torsion?

Key findings

• Main Theorem 1.1 establishes a unique extension of any quasismooth hyperelliptic K3 surface T ⊂ ℙ(2⁴,3⁴,4) with 10×1/2 points to a quasismooth Fano 6-fold W ⊂ ℙ(1⁴,2⁴,3⁴,4) with the same singularities.
• Corollary 1.2 constructs a 15-parameter family of hyperelliptic surfaces Y of general type with p_g=1, q=0, K²=2, and trivial torsion, each a complete intersection of type (1,1,1,2) in W.
• Theorem 1.3 proves the existence of an 8-parameter family of Godeaux surfaces X with ℤ/2 torsion, obtained as ℤ/2 quotients of surfaces Y from Corollary 1.2.
• The Godeaux surfaces constructed have p_g=0, K²=1, and torsion group ℤ/2, with the moduli space expected to be irreducible and 8-dimensional.
• The involution on the Fano 6-fold W is shown to lift from the K3 surface T and act with exactly four isolated fixed points of type 1/2, ensuring the quotient is well-behaved.
• The construction provides a new geometric method to access Godeaux surfaces with ℤ/2 torsion, potentially yielding an irreducible 8-dimensional component in the moduli space.