Autoduality of compactified Jacobians for curves with plane singularities

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an integral projective curve with at most planar singularities. Consider its Jacobian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the compactified Jacobian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J overbar"> <mml:semantics> <mml:mover> <mml:mi>J</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex">\overline {J}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We construct a flat family <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P overbar"> <mml:semantics> <mml:mover> <mml:mi>P</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex">\overline {P}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Cohen-Macaulay sheaves on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J overbar"> <mml:semantics> <mml:mover> <mml:mi>J</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex">\overline {J}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> parametrized by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J overbar"> <mml:semantics> <mml:mover> <mml:mi>J</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex">\overline {J}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; its restriction to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J times upper J overbar"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo> × </mml:mo> <mml:mover> <mml:mi>J</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">J\times \overline {J}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Poincaré line bundle. We prove that the Fourier-Mukai transform given by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P overbar"> <mml:semantics> <mml:mover> <mml:mi>P</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex">\overline {P}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an auto-equivalence of the derived category of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J overbar"> <mml:semantics> <mml:mover> <mml:mi>J</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex">\overline {J}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .