Algebraic growth of the Cremona group

We initiate the study of the “algebraic growth” of groups of automorphisms and birational transformations of algebraic varieties. Our main result concerns <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="sans-serif">Bir</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Bir(P²) , the Cremona group in 2 variables. This group is the union, for all degrees <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> d≥ 1 , of the algebraic variety <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="sans-serif">Bir</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>d</m:mi> </m:msub> </m:mrow> </m:math> Bir(P²)d of birational transformations of the plane of degree 𝑑. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>N</m:mi> <m:mi>d</m:mi> </m:msub> </m:math> Nd denote the number of irreducible components of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="sans-serif">Bir</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>d</m:mi> </m:msub> </m:mrow> </m:math> Bir(P²)d . We describe the asymptotic growth of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>N</m:mi> <m:mi>d</m:mi> </m:msub> </m:math> Nd as 𝑑 goes to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>+</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> +∞ , showing that there are two constants 𝐴 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>B</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> B>0 such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>A</m:mi> <m:mo>⁢</m:mo> <m:msqrt> <m:mrow> <m:mi>ln</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msqrt> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:mi>ln</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo maxsize="160%" minsize="160%">(</m:mo> <m:mrow> <m:mi>ln</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo maxsize="160%" minsize="160%">(</m:mo> <m:mrow> <m:munder> <m:mo lspace="0em" movablelimits="false">∑</m:mo> <m:mrow> <m:mi>e</m:mi> <m:mo>≤</m:mo> <m:mi>d</m:mi> </m:mrow> </m:munder> <m:msub> <m:mi>N</m:mi> <m:mi>e</m:mi> </m:msub> </m:mrow> <m:mo maxsize="160%" minsize="160%">)</m:mo>