Birational morphisms of the plane (2004)

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A squared"> <mml:semantics> <mml:msup> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">A^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the affine plane over a field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Birational morphisms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A squared"> <mml:semantics> <mml:msup> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">A^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are mappings <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A squared right-arrow upper A squared"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false"> → </mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">A^2 \to A^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> given by polynomial mappings <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi> φ </mml:mi> <mml:annotation encoding="application/x-tex">\varphi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the polynomial algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-bracket x comma y right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">K[x,y]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that for the quotient fields, one has <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis phi left-parenthesis x right-parenthesis comma phi left-parenthesis y right-parenthesis right-parenthesis equals upper K left-parenthesis x comma y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> φ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi> φ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">K(\varphi (x), \varphi (y)) = K(x,y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau Subscript x"> <mml:semantics> <mml:msub> <mml:mi> τ </mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\tau _x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> given by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow x comma y right-arrow x y"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>y</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>x</mml:mi> <mml:mi>y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x \to x, ~y \to xy</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau Subscript x"> <mml:semantics> <mml:msub> <mml:mi> τ </mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\tau _x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This question was answered in the negative by P. Russell (in an informal communication). In this paper, we give a simple combinatorial solution of the same problem. More importantly, our method yields an algorithm for deciding whether a given birational morphism can be factored that way.