Compactification of the moduli space of hyperplane arrangements

Abstract

Consider the moduli space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Superscript 0"> <mml:semantics> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">M^0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of arrangements of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> hyperplanes in general position in projective <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis r minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(r-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-space. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r equals 2"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">r=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the space has a compactification given by the moduli space of stable curves of genus <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> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> marked points. In higher dimensions, the analogue of the moduli space of stable curves is the moduli space of stable pairs: pairs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper S comma upper B right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(S,B)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consisting of a variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (possibly reducible) and a divisor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B equals upper B 1 plus midline-horizontal-ellipsis plus upper B Subscript n"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">B=B_1+\dots +B_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, satisfying various additional conditions. We identify the closure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Superscript 0"> <mml:semantics> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">M^0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the moduli space of stable pairs as Kapranov’s Hilbert quotient compactification of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Superscript 0"> <mml:semantics> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">M^0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and give an explicit description of the pairs at the boundary. We also construct additional irreducible components of the moduli space of stable pairs.