[Paper Review] Bi-canonical representations of finite automorphisms acting on Enriques surfaces

Hisanori Ohashi|arXiv (Cornell University)|Apr 3, 2015

Advanced Algebra and Geometry3 references3 citations

TL;DR

This paper classifies finite non-semi-symplectic automorphisms on Enriques surfaces by their bi-canonical representations, proving that only three types exist: (order 4, index 2), (order 8, index 4), and (order 8, index 2). Using Horikawa's birational models and K3 covers, it constructs explicit examples and establishes uniqueness theorems for maximal-dimensional moduli families, showing maximal moduli dimensions of 5, 2, and 2 respectively.

ABSTRACT

We classify the bi-canonical representations of finite automorphisms on Enriques surfaces. There are three types of non-trivial cases and examples are given explicitly by Horikawa models. In particular, finite non-semi-symplectic automorphisms exist only in orders 4 and 8. One corollary is that any finite cyclic subgroup in the automorphism group of an Enriques surface has order 1, 2, 3, 4, 5, 6, 8. Moreover, for two of the three types, a uniqueness theorem for maximal-dimensional families is given.

Motivation & Objective

• To classify finite non-semi-symplectic automorphisms on Enriques surfaces via their bi-canonical representations.
• To determine the possible orders and indices of such automorphisms, particularly in the non-semi-symplectic case.
• To construct explicit examples using Horikawa's birational models of elliptic Enriques surfaces doubly covering P¹×P¹.
• To establish uniqueness theorems for maximal-dimensional families of automorphisms in two of the three types.
• To determine the maximal possible moduli dimension for families of such automorphisms.

Proposed method

• Uses Horikawa's birational projective model of elliptic Enriques surfaces doubly covering P¹×P¹ to construct explicit equations invariant under automorphisms of order 4 and 8.
• Analyzes the action of automorphisms on the space of global bi-2-forms H⁰(OS(2KS)) to determine the index I(σ) of the automorphism σ.
• Lifts automorphisms from the Enriques surface S to its K3 cover X, using the action on H⁰(OX(KX)) to define the index and order.
• Applies the holomorphic Lefschetz fixed-point theorem to analyze fixed-point sets and derive lattice isomorphisms, such as H²(X,Z)^φ² ≅ U(2).
• Uses the Néron-Severi lattice and quotient geometry to show that X/φ² ≅ P¹×P¹, and determines the action of the residue group K₄ ≅ (Z/2)² on the quotient.
• Classifies subgroups of PGL(2,C)×PGL(2,C) acting on P¹×P¹, showing that only specific normal forms arise, leading to uniqueness up to isomorphism.

Experimental results

Research questions

• RQ1 What are the possible orders and indices of finite non-semi-symplectic automorphisms on Enriques surfaces?
• RQ2 Can explicit families of such automorphisms be constructed using Horikawa models?
• RQ3 What is the maximal possible dimension of a moduli family for such automorphisms?
• RQ4 Are there uniqueness results for maximal-dimensional families of non-semi-symplectic automorphisms?
• RQ5 How do the automorphism groups on the K3 cover relate to the geometry of the quotient surface?

Key findings

• Finite non-semi-symplectic automorphisms on Enriques surfaces exist only in orders 4 and 8, with indices 2 or 4.
• The only possible bi-canonical representation types are (4,2), (8,4), and (8,2), as proven in Theorem 1.4.
• For the (4,2) case, the maximal moduli dimension is 5, and Example 1.1 realizes this bound.
• For the (8,4) and (8,2) cases, the maximal moduli dimension is 2, and Examples 1.2 and 1.3 achieve these bounds.
• The (4,2) case admits a uniqueness theorem for maximal-dimensional families, as shown via the classification of K₄ actions on P¹×P¹.
• The K3 cover X of an Enriques surface with such automorphisms has Picard number 10, and the fixed lattice H²(X,Z)^φ² is isomorphic to U(2).