Nonabelian Jacobian J(X; L, d): Main Properties
In this section we introduce the main objects of our study and recall the main results of [R1].
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Nonabelian Jacobian J.X I L; d/: Main Properties
In this section we introduce the main objects of our study and recall the main results of [R1].
2.1 Construction of J.X I L; d/ By analogy with the classical Jacobian of a smooth projective curve our Jacobian is supposed to be the parameter space of a certain distinguished family of torsion free sheaves of rank 2 over X , having the Chern invariants .L; d /. Its formal definition is as follows. One starts with the Hilbert scheme X Œd of closed zero-dimensional subschemes of X having length d . Over X Œd there is the universal scheme Z of such subschemes Z p1
X
X X Œd
(2.1)
p2
X Œd
where pi .i D 1; 2/ is the restriction to Z of the projections pri ; i D 1; 2; of the Cartesian product X X Œd onto the corresponding factor. For a point 2 X Œd , the fibre p2 ./ is isomorphic via p1 with the subscheme Z of X corresponding to , i.e. Z D p1 .p2 .//: (2.2) In the sequel we often make no distinction between Z and the fibre p2 ./ itself. If Z is a closed subscheme of dimension zero and length d , then ŒZ will denote the corresponding point in the Hilbert scheme X Œd .
I. Reider, Nonabelian Jacobian of Projective Surfaces, Lecture Notes in Mathematics 2072, DOI 10.1007/978-3-642-35662-9 2, © Springer-Verlag Berlin Heidelberg 2013
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2 Nonabelian Jacobian J.XI L; d /: Main Properties
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The next step is to fix a line bundle OX .L/ corresponding to a divisor L on X . It will be assumed throughout the paper that OX .L/ satisfies the following condition H 0 .OX .L// D H 1 .OX .L// D 0:
(2.3)
We are aiming at geometric applications, where the divisor L will be sufficiently positive (e.g. L is ample), so the above condition is quite natural. Once a divisor L and a positive integer d are fixed we consider the following morphism of sheaves on X Œd
H 0 .OX .L C KX // ˝ OX Œd We define
p2 p1 OX .L C KX / :
J.X I L; d / WD Proj.S coker/;
(2.4)
(2.5)
where S coker is the symmetric algebra of coker. By definition J.X I L; d / comes with the natural projection W J.X I L; d / ! X Œd
(2.6)
and the invertible sheaf OJ.X IL;d / .1/ such that the direct image OJ.X IL;d / .1/ D coker
(2.7)
(when X; L and d are fixed and no ambiguity is likely, we will omit these parameters in the notation for the Jacobian and simply write J instead of J.X I L; d /). Observe that the set of closed points of the fibre of over a point ŒZ in X Œd is naturally homeomorphic to the projective space P.H 1 .IZ .L C KX // /. By Serre duality on X H 1 .IZ .L C KX // D Ext 1 .IZ .L C KX /; O.KX // D Ext 1 .IZ .L/; OX /: (2.8) To simplify the notations the last space will be denoted by ExtZ1 throughout the paper. Thus the set of closed points of J.X I L; d / is in one to one correspondence with the set of pairs .ŒZ; Œ˛/, where ŒZ 2 X Œd and Œ˛ 2 P.ExtZ1 /. Alternatively, a pair .ŒZ; Œ˛/ can be thought of as the pair .E; Œe/, where E is the torsion free sheaf sitting in the middle of the extension sequence defined by the clas
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