[Paper Review] Autoduality of compactified Jacobians for curves with plane singularities

D. Arinkin|arXiv (Cornell University)|Jan 21, 2010

Algebraic Geometry and Number Theory16 references8 citations

TL;DR

This paper establishes autoduality for compactified Jacobians of integral projective curves with planar singularities by constructing a universal Cohen-Macaulay sheaf ${\overline{P}}$ on ${\overline{J}} \times {\overline{J}}$ that extends the Poincaré line bundle from the Jacobian $J$. The Fourier-Mukai transform associated to ${\overline{P}}$ is shown to be an equivalence of derived categories, proving that ${\overline{J}}$ is self-dual in the derived sense.

ABSTRACT

Let C be an integral projective curve with planar singularities. Consider its Jacobian J and the compactified Jacobian J'. We construct a flat family P of Cohen-Macaulay sheaves on J' parametrized by J'; over J, the family P is the Poincare line bundle. We prove that the Fourier-Mukai transform given by P is an auto-equivalence of the derived category of J'.

Motivation & Objective

• To establish derived autoduality for compactified Jacobians of integral projective curves with planar singularities.
• To construct a universal family of Cohen-Macaulay sheaves on ${\overline{J}} \times {\overline{J}}$ extending the Poincaré bundle from $J \times {\overline{J}}$.
• To prove that the Fourier-Mukai transform associated to this sheaf is an equivalence of derived categories on ${\overline{J}}$.
• To show that ${\overline{J}}$ is isomorphic to a connected component of the moduli space of torsion-free sheaves of rank one on itself.

Proposed method

• Construct a coherent sheaf ${\overline{P}}$ on ${\overline{J}} \times {\overline{J}}$ that restricts to the Poincaré line bundle on $J \times {\overline{J}} \cup {\overline{J}} \times J$.
• Prove that ${\overline{P}}$ is flat over each factor of ${\overline{J}} \times {\overline{J}}$, making it a family of sheaves parametrized by ${\overline{J}}$.
• Use the universal property of ${\overline{P}}$ to identify ${\overline{J}}$ with a connected component of the moduli space of torsion-free sheaves of rank one on ${\overline{J}}$.
• Apply the Fourier-Mukai transform $\mathfrak{F}(\mathcal{G}) = Rp_{2,*}(p_1^*\mathcal{G} \otimes {\overline{P}})$ and prove it is an equivalence on $D^b({\overline{J}})$.
• Leverage results from [5] and [14], especially the fact that $P$ extends to a universal family on $J \times {\overline{J}} \cup {\overline{J}} \times J$, to build the extension ${\overline{P}}$.
• Use Serre duality and semicontinuity of Hilbert polynomials to show that the transform of a sheaf $M$ supported at a point corresponds to a skyscraper sheaf, proving the inverse map is algebraic.

Experimental results

Research questions

• RQ1 Can the Poincaré line bundle on $J \times {\overline{J}}$ be extended to a universal sheaf on ${\overline{J}} \times {\overline{J}}$ for curves with planar singularities?
• RQ2 Is the Fourier-Mukai transform associated to such an extension an equivalence of derived categories on ${\overline{J}}$?
• RQ3 Does ${\overline{J}}$ admit a self-duality in the derived category sense, generalizing the classical autoduality of smooth Jacobians?
• RQ4 Can the compactified Jacobian ${\overline{J}}$ be identified with a connected component of the moduli space of torsion-free sheaves of rank one on itself?
• RQ5 What is the relationship between the derived autoduality of ${\overline{J}}$ and the Hitchin fibration for $\mathrm{GL}(n)$?

Key findings

• A flat family of Cohen-Macaulay sheaves ${\overline{P}}$ on ${\overline{J}} \times {\overline{J}}$ is constructed, extending the Poincaré bundle from $J \times {\overline{J}} \cup {\overline{J}} \times J$.
• The restriction of ${\overline{P}}$ to $J \times {\overline{J}}$ agrees with the standard Poincaré line bundle, and the formula $P_{(L,F)} = \det R\Gamma(L \otimes F) \otimes \det R\Gamma(O_C) \otimes \det R\Gamma(L)^{-1} \otimes \det R\Gamma(F)^{-1}$ holds for $L,F \in J$.
• The Fourier-Mukai transform $\mathfrak{F}$ defined by ${\overline{P}}$ is an equivalence of derived categories $D^b({\overline{J}}) \to D^b({\overline{J}})$.
• The map $\rho: {\overline{J}} \to \mathop{\mathrm{Pic}}\nolimits({\overline{J}})^{=}$ sending $F \in {\overline{J}}$ to ${\overline{P}}_F$ identifies ${\overline{J}}$ with a connected component of the moduli space of torsion-free sheaves of rank one on ${\overline{J}}$.
• For any $M \in \mathop{\mathrm{Pic}}\nolimits({\overline{J}})^{=}$ satisfying $h^i(\mathfrak{F}(M)) = \delta_{i,g}$, the transform $\mathfrak{F}(M)$ is isomorphic to $\mathcal{O}_{F^\vee}[-g]$, proving the inverse map is algebraic.
• The universal sheaf $\Psi_{\mathrm{univ}}[g]$ is shown to be isomorphic to $\mathcal{O}_\Delta \otimes \pi^* \det(\mathfrak{j})$, confirming the sheaf is Cohen-Macaulay and supports the derived equivalence.