[Paper Review] Sarkisov program for generalized pairs

Jihao Liu|arXiv (Cornell University)|Feb 12, 2018

Algebraic Geometry and Number Theory11 references4 citations

TL;DR

This paper establishes the Sarkisov program for $\mathbb{Q}$-factorial gklt generalized pairs, proving that any two birational Mori fiber spaces arising from a common generalized pair via a $K_W + B_W + M_W$-MMP are connected by a finite sequence of Sarkisov links. The result extends the classical Sarkisov program to the generalized pair setting, ensuring birational maps between Mori fiber spaces decompose into standard links, preserving generalized log-canonical singularities.

ABSTRACT

In this paper we show that any two birational Mori fiber spaces of $\Qq$-factorial gklt g-pairs are connected by a finite sequence of Sarkisov links.

Motivation & Objective

• To extend the Sarkisov program—previously established for klt pairs—to the broader category of generalized pairs.
• To prove that birational maps between Mori fiber spaces arising from a common generalized pair are composed of finitely many Sarkisov links.
• To preserve generalized log-canonical (gklt) singularities throughout the decomposition, ensuring birational invariance of the singularities under the links.
• To generalize the minimal model program framework to include generalized pairs with nef moduli divisors, particularly in the non-pseudo-effective case.

Proposed method

• Construct a Sarkisov program with double scaling using two ample $\mathbb{R}$-divisors $L_W$ and $H_W$ on $W$ to control the MMP steps.
• Use the existence of a common log resolution to assume $W$ is g-terminal and $\rho_X, \rho_Y$ are morphisms, simplifying the birational geometry.
• Define a sequence of birational maps $X_0 \dashrightarrow X_1 \dashrightarrow \cdots \dashrightarrow X_n$ via scaling of $(L_W, H_W)$, ensuring each step is a Sarkisov link.
• Prove that the final map $f: X \dashrightarrow Y$ is an isomorphism in codimension one by showing $X_n \cong Y$ via log canonical models of a perturbed pair $(W, \Delta_W)$.
• Use numerical positivity and nefness of $K_{X_n} + B_n + h_n H_n + M_{X_n}$ to derive a contradiction if $h_n < 1$, forcing $h_n = 1$ and $X_n \cong Y$.
• Apply the canonical bundle formula and adjunction to show that the final Mori fiber space structure is preserved under the link decomposition.

Experimental results

Research questions

• RQ1 Can the Sarkisov program be extended from klt pairs to generalized pairs with nef moduli divisors?
• RQ2 Are two birational Mori fiber spaces obtained from the same generalized pair connected by a finite sequence of Sarkisov links?
• RQ3 Does the generalized log-canonical (gklt) property persist through the Sarkisov link decomposition?
• RQ4 Is the birational map between two Mori fiber spaces arising from a common generalized pair decomposable into standard links with double scaling?
• RQ5 Can the minimal model program for generalized pairs be used to construct a full Sarkisov program with controlled singularities?

Key findings

• Any two birational Mori fiber spaces associated to a $\mathbb{Q}$-factorial gklt generalized pair via a $K_W + B_W + M_W$-MMP are connected by a finite sequence of Sarkisov links.
• The induced birational map $f: X \dashrightarrow Y$ is a composition of Sarkisov links with double scaling, as defined in Construction 4.3.
• The final map satisfies $X_n \cong Y$, meaning the decomposition terminates in an isomorphism in codimension one.
• The generalized log-canonical (gklt) property is preserved along the entire sequence: if the initial pair is generalized $\epsilon$-lc, so are all intermediate pairs.
• The proof relies on contradiction via numerical positivity: if the scaling parameter $h_n < 1$, then $K_W + B_W + h_n H_W + M_W$ would be pseudo-effective, contradicting the non-pseudo-effectiveness of $K_W + B_W + M_W$.
• The isomorphism $X_n \cong Y$ is established via log canonical models of a perturbed pair $(W, \Delta_W)$, where $\Delta_W$ is a general perturbation of $B_W + H_W + M_W$.