Extendability of the $$B_2$$-Arrangement

Abstract

<jats:p> Let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$({\mathscr {A}},m)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be a free multiarrangement, and let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${\mathscr {E}}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> be an extension of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$({\mathscr {A}},m)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . It is well known that if <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${\mathscr {A}}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> is the Coxeter arrangement of type <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$A_2$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> , then a free extension of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$({\mathscr {A}},m)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> always exists. In this work, we demonstrate that if <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${\mathscr {A}}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> is the Coxeter arrangement of type <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$B_2$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> , there exist infinitely many multiplicities for which no free extension of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$({\mathscr {A}},m)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> exists. This result has immediate consequences for the existence of free extensions in higher rank. </jats:p>