[Paper Review] A construction of numerical Campedelli Surfaces with \Z/6 torsion group

Jorge Neves, Stavros Argyrios Papadakis|ArXiv.org|Jul 2, 2007

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper constructs the first known example of a numerical Campedelli surface with algebraic fundamental group ℤ/6ℤ using serial unprojection of Kustin–Miller type to build the canonical ring of an étale six-to-one cover. The method involves constructing a Gorenstein ring via generic Pfaffian ideals and proving the existence of a nonsingular surface with pg=5, K²=12, and a free ℤ/6ℤ action, thereby resolving the existence question for this torsion group.

ABSTRACT

We produce a family of numerical Campedelli surfaces with \Z/6 torsion by constructing the (Gorenstein codimension 5) canonical ring of the étale six to one cover using serial unprojection. In Section 2 we develop the necessary algebraic machinery. Section 3 contains the numerical Campedelli surface construction, while Section 4 contains remarks and open questions.

Motivation & Objective

• To resolve the existence question for numerical Campedelli surfaces with algebraic fundamental group ℤ/6ℤ, which was previously unknown.
• To construct a canonical ring for an étale six-to-one cover of a numerical Campedelli surface using advanced algebraic geometry techniques.
• To demonstrate that such surfaces exist with pg=5, K²=12, and a free ℤ/6ℤ action via explicit unprojection methods.
• To provide a framework using serial unprojection and Hilbert series that may generalize to other torsion groups like ℤ/2ℤ and ℤ/3ℤ.

Proposed method

• The authors define a generic $\binom{n}{2}$ Pfaffians ideal in a polynomial ring with variables $x_i, z_i, y_i, r_{d_1\cdots d_n}$, graded to ensure Gorenstein structure.
• They prove that this ideal is Gorenstein of codimension $n+1$, generalizing earlier results by Frantzen and others.
• For $n=4$, they apply the construction to realize a 3-fold $V$ in $\mathbb{P}(1^5,2^4)$ with the correct Hilbert series, using four Kustin–Miller unprojections from a degree 4 hypersurface in $\mathbb{P}(1^5)$.
• They construct a 6-fold in $\mathbb{P}(1^8,2^4)$ by unprojecting four linear subspaces of dimension 5, then take three linear and one quadratic sections to obtain a surface with $p_g=5$, $K^2=12$, and a $\mathbb{Z}/6$-action.
• Using character theory and $G$-Hilbert series, they identify a subfamily with a good $\mathbb{Z}/6$-action, ensuring the quotient is a numerical Campedelli surface.
• They verify the absence of fixed points under the $\mathbb{Z}/6$-action via character analysis and contradiction arguments on the eigenspace decomposition of the action.

Experimental results

Research questions

• RQ1 Does there exist a numerical Campedelli surface with algebraic fundamental group ℤ/6ℤ, given that no such examples were previously known?
• RQ2 Can the canonical ring of an étale six-to-one cover of such a surface be constructed via serial unprojection of Kustin–Miller type?
• RQ3 Is the generic $\binom{n}{2}$ Pfaffians ideal Gorenstein of codimension $n+1$ for general $n$?
• RQ4 Can the unprojection framework be extended to construct other numerical Campedelli surfaces with different torsion groups, such as ℤ/2ℤ or ℤ/3ℤ?
• RQ5 What is the dimension of the family of ℤ/6ℤ-torsion numerical Campedelli surfaces constructed in this work?

Key findings

• The paper constructs a nonsingular regular surface with $p_g=5$, $K^2=12$, and a free $\mathbb{Z}/6$-action, proving its existence as a canonical cover.
• The generic $\binom{n}{2}$ Pfaffians ideal is shown to be Gorenstein of codimension $n+1$, generalizing earlier results for $n=4$.
• The étale six-to-one cover is realized as a quadratic section of a Fano threefold $V$ in $\mathbb{P}(1^5,2^4)$, constructed via four successive Kustin–Miller unprojections.
• The $\mathbb{Z}/6$-action is shown to be basepoint-free and fixed-point-free through character-theoretic analysis and contradiction arguments on eigenspace components.
• The construction confirms the existence of numerical Campedelli surfaces with $\pi_1^{\text{alg}} = \mathbb{Z}/6\mathbb{Z}$, closing a gap in the classification of such surfaces.
• The authors expect that similar unprojection techniques could yield constructions for $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$ torsion cases, based on candidate Fano threefolds in Brown’s database.