[Paper Review] On degrees of birational mappings

Serge Cantat, Junyi Xie|arXiv (Cornell University)|Feb 23, 2018

Mathematical Dynamics and Fractals12 references3 citations

TL;DR

This paper establishes fundamental properties of degree sequences for birational maps on projective varieties, proving that the degree sequence is bounded if and only if it is bounded along an infinite subsequence, and showing that unbounded degree growth cannot be arbitrarily slow—specifically, it is asymptotically bounded below by the inverse of the diagonal Ackermann function in the case of projective space. The results rely on p-adic methods and geometric degree analysis via projective compactifications and divisor theory.

ABSTRACT

We prove that the degrees of the iterates ${ m deg}(f^n)$ of a birational map satisfy $\liminf({ m deg}(f^n))

Motivation & Objective

• To characterize when the degree sequence of a birational map is bounded, particularly in relation to subsequences.
• To determine the minimal possible growth rate of unbounded degree sequences for birational maps on projective varieties.
• To establish that degree sequences cannot grow arbitrarily slowly unless they are bounded, using geometric and p-adic techniques.
• To unify and extend degree-theoretic tools for birational dynamics, especially in higher dimensions where such sequences are poorly understood.
• To prove equivalence between different notions of degree (via divisor theory and intersection theory) in the context of automorphisms and birational maps.

Proposed method

• Introduces a new degree function $\deg^H(f)$ using divisor theory on a normal projective compactification $Y$ of an affine variety $X$, defined via the minimal $d$ such that $ (f^*P) + dH \geq 0 $ for all $P \in A_1 $.
• Uses Riemann-Roch to bound $\dim A_d \leq \gamma d^k$, leading to a polynomial upper bound on the number of endomorphisms of degree $\leq d$.
• Applies a $p$-adic method via the structure of the space of piecewise linear subsets in the moduli space of divisors, tracking the decay of sets $Z_D'(i)$ under iteration.
• Employs Siu’s inequality and birational geometry to relate $\deg^H(f)$ to the standard intersection-theoretic degree $\deg_H(f)$, proving $ \frac{1}{k}\deg^H(f) \leq \frac{1}{(H^k)}\deg_H(f) \leq \deg^H(f) $.
• Analyzes the dynamics of iterated pullbacks via the structure of the group $G$ of birational maps of bounded degree, estimating component counts in fibered intersections.
• Uses induction on the number of irreducible components of piecewise linear sets to bound the length $\ell(D)$ of the decreasing chain $Z_D'(i)$, leading to a bound in terms of the Ackermann function inverse.

Experimental results

Research questions

• RQ1 Under what conditions is the degree sequence $\deg(f^n)$ bounded for a birational map $f$?
• RQ2 Can the growth of $\deg(f^n)$ be arbitrarily slow if the sequence is unbounded?
• RQ3 What is the minimal possible growth rate of $\deg(f^n)$ for unbounded sequences, particularly on $\mathbb{P}^k$?
• RQ4 How do different notions of degree (e.g., $\deg^H(f)$ and $\deg_H(f)$) relate in the context of automorphisms and birational maps?
• RQ5 To what extent can the dynamics of degree sequences be controlled via geometric and arithmetic methods, such as $p$-adic analysis or piecewise linear geometry?

Key findings

• The degree sequence $\deg(f^n)$ is bounded if and only if it is bounded along some infinite subsequence, resolving a key question about subsequential boundedness.
• If $\deg(f^n)$ is unbounded, then $\max_{0\leq j\leq n} \deg(f^j)$ grows at least as fast as the inverse of the diagonal Ackermann function, meaning growth cannot be arbitrarily slow.
• For automorphisms of affine varieties, the number of $n$ with $\deg(f^n) \leq d$ is at most $\alpha d^k$, so $\max_{0\leq j\leq n} \deg(f^j) \geq (n/\alpha)^{1/k}$, showing polynomial lower bounds on growth.
• The degree $\deg^H(f)$ defined via divisor theory satisfies $ \frac{1}{k}\deg^H(f) \leq \frac{1}{(H^k)}\deg_H(f) \leq \deg^H(f) $, proving equivalence with the standard intersection-theoretic degree.
• The length $\ell(D)$ of the decreasing chain of piecewise linear sets $Z_D'(i)$ is bounded by $ S(\binom{k+D}{k} - 2) + 1 $, where $S$ is a function related to the Ackermann function, yielding the slowest possible growth rate.
• The growth of $\deg(f^n)$ on $\mathbb{P}^k$ cannot be slower than the inverse of the diagonal Ackermann function, establishing a sharp lower bound on the rate of divergence for unbounded sequences.