[Paper Review] Variations on Log Sarkisov Program for Surfaces

Adrien Dubouloz, Stéphane Lamy|ArXiv.org|Feb 18, 2008

Geometric Analysis and Curvature Flows13 references3 citations

TL;DR

This paper presents a log Sarkisov program for smooth quasi-projective surfaces with an irreducible boundary divisor, factorizing automorphisms into elementary links where all blow-ups and contractions occur on the boundary. The key contribution is a presentation-like description of the automorphism group via such factorizations, valid when the compactification is smooth.

ABSTRACT

Let (S, BS) be the log-pair associated with a compactification of a given smooth quasi-projective surface V . Under the assumption that the boundary BS is irreducible, we propose an algorithm, in the spirit of the (log) Sarkisov program, to factorize any automorphism of V into a sequence of elementary links in the framework of the logarithmic Mori theory. The new noteworthy feature of our algorithm is that all the blow-ups and contractions involved in the process occur on the boundary.

Motivation & Objective

• To factorize automorphisms of a smooth quasi-projective surface V that do not extend to biregular automorphisms on a compactification S.
• To develop an algorithm in the spirit of the log Sarkisov program for surfaces with irreducible boundary divisor B_S.
• To describe the automorphism group of V in a way analogous to a presentation by generators and relations.
• To ensure all blow-ups and contractions in the factorization process occur on the boundary divisor B_S.
• To extend the log Sarkisov framework to cases where the boundary has coefficient 1, overcoming limitations of standard klt/dlt assumptions.

Proposed method

• The algorithm uses log Mori theory to decompose birational maps between compactifications of V into elementary links.
• All blow-ups and contractions are performed exclusively on the boundary divisor B_S, ensuring control over the geometry of the compactification.
• The method relies on resolving base points of birational maps via sequences of blow-ups centered at points on the boundary.
• It employs ramification formulas for the canonical divisor K_X, the boundary divisor B_X, and the strict transform of an ample divisor H_Y.
• The maximal multiplicity λ is defined as the maximum of m_i / (c_i - b_i), where m_i, c_i, b_i are coefficients from the pullbacks of H_Y, K_Y, and B_Y.
• The process constructs intermediate surfaces S_i via sequences of blow-ups and contractions, with factorizations expressed as chains of elementary links.

Experimental results

Research questions

• RQ1 Can automorphisms of a smooth quasi-projective surface V be decomposed into elementary links within the log Mori framework when the boundary divisor is irreducible?
• RQ2 What is the structure of the automorphism group of V when the compactification S is smooth and the boundary B_S is irreducible?
• RQ3 How can the log Sarkisov program be adapted to cases where the boundary divisor has coefficient 1, violating standard klt/dlt conditions?
• RQ4 To what extent can the blow-ups and contractions involved in the factorization be restricted to the boundary divisor B_S?
• RQ5 Can such a factorization yield a presentation of the automorphism group analogous to generators and relations?

Key findings

• The automorphism group of V admits a presentation-like structure via a sequence of elementary links, with all modifications occurring on the boundary divisor B_S.
• For a smooth compactification S, the automorphism group of V is described through a factorization into elementary links that are fully controlled by the boundary geometry.
• The factorization of a specific automorphism h: V → V is shown to consist of six elementary links: S₀ ↔ S₁ ↔ S₂ ↔ S₃ ↔ S₂ ↔ S₁ ↔ S₀.
• The intermediate surface S₃ is obtained from the weighted projective plane P²(2) via a sequence of blow-ups and contractions similar to those used to construct S₂ from P².
• The resolution of the automorphism h₀: S₀ ⇢ S₀ involves a minimal log resolution with a boundary structure visualized in Figure 13, featuring exceptional divisors with self-intersection numbers -1, -2, -4, -2, -4, -2, -1.
• The factorization of the map σ: S₀ ⇢ S₂ consists of two elementary links, and the same applies to the inverse map σ⁻¹, with intermediate surfaces S₁ having singularities of type A₃,₂ and A₂,₁.