[Paper Review] Int-amplified endomorphisms on normal projective surfaces

Yohsuke Matsuzawa, Shou Yoshikawa|arXiv (Cornell University)|Feb 16, 2019

Algebraic Geometry and Number Theory11 references4 citations

TL;DR

This paper classifies normal projective surfaces admitting int-amplified endomorphisms by constructing an equivariant minimal model program (MMP) and showing the output is either a Q-abelian surface, a quasi-étale quotient of a smooth surface, a Mori dream space, or a projective cone over an elliptic curve. The key contribution is the explicit construction of quasi-étale covers in cases where the anti-canonical divisor has Iitaka dimension 0 or 1, revealing new geometric structures compatible with endomorphisms.

ABSTRACT

We investigate int-amplified endomorphisms on normal projective surfaces. We prove that the output of the equivariant MMP is either a Q-abelian surface, a (equivariant) quasi-étale quotient of a smooth projective surface, a Mori dream space, or a projective cone of an elliptic curve.

Motivation & Objective

• To classify normal projective surfaces that admit int-amplified endomorphisms over an algebraically closed field of characteristic zero.
• To understand the geometric structure of such surfaces by applying the minimal model program (MMP) in an equivariant way.
• To construct explicit quasi-étale finite covers for surfaces where the anti-canonical divisor has Iitaka dimension 0 or 1.
• To identify the possible outcomes of the equivariant MMP for int-amplified endomorphisms on surfaces, including Mori dream spaces and projective cones.

Proposed method

• Apply the equivariant minimal model program (MMP) to a normal projective surface X with an int-amplified endomorphism f, ensuring f^n induces endomorphisms on each step of the sequence.
• Use the property that f^*H - H is ample for some ample Cartier divisor H to ensure compatibility with the MMP and control singularities.
• Construct quasi-étale finite covers h: Y → X from smooth surfaces Y that are minimal ruled surfaces over elliptic curves, lifting the endomorphism f to Y.
• Utilize the geometry of ruled surfaces over elliptic curves and the action of multiplication-by-n maps on the base to build explicit examples of int-amplified endomorphisms.
• Analyze the Iitaka dimension of the anti-canonical divisor -K_X to classify the possible outcomes of the MMP, distinguishing cases by κ(-K_X) = 0, 1, 2, or pseudo-effectiveness.
• Leverage the fact that the induced endomorphism on the base curve (P^1 or point) lifts to the cover, ensuring equivariance and compatibility with the quotient structure.

Experimental results

Research questions

• RQ1 What are the possible geometric structures of a normal projective surface X that admits an int-amplified endomorphism?
• RQ2 Can the equivariant MMP for such surfaces be used to classify the possible outcomes, and what are the invariants that distinguish them?
• RQ3 In cases where the anti-canonical divisor has Iitaka dimension 0 or 1, can one construct a quasi-étale cover from a smooth surface that lifts the endomorphism?
• RQ4 How does the structure of the surface change depending on whether the canonical divisor is pseudo-effective or not?
• RQ5 What are the precise conditions under which the surface is a Mori dream space or a projective cone over an elliptic curve?

Key findings

• The surface X is Q-Gorenstein log canonical and admits an f^n-equivariant MMP sequence ending in a curve C or a point.
• If K_X is pseudo-effective, the MMP ends with a Fano contraction to a curve C; otherwise, it ends in a point with Picard number one and -K_X ample.
• In the case where C ≃ P^1 and κ(-K_X) = 0 or 1, there exists a quasi-étale finite cover h: Y → X from a smooth minimal ruled surface over an elliptic curve, with f lifting to an endomorphism on Y.
• When κ(-K_X) = 2, X is a Mori dream space, and when the MMP ends in a point, X is either a projective cone over an elliptic curve or a Mori dream space.
• The construction of the quasi-étale cover in cases (3) and (4) of Theorem 1.2 provides a new geometric realization of surfaces with int-amplified endomorphisms via ruled surfaces over elliptic curves.
• Explicit examples are constructed using projective bundles over elliptic curves and the action of multiplication-by-n maps, yielding int-amplified endomorphisms on the quotient surfaces.