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Jacobians and differents of projective varieties. (English) Zbl 0608.14033

Let X be a reduced and irreducible projective variety of dimension \(d\) over a perfect field k. Our main result is the equality of \({\mathcal O}_ X\)-modules: (*) \(J_ X\quad \phi (\pi ^ *\Omega ^ d_{X/k})=\theta _ k(X| Y)\quad \phi (\Omega ^ d_{X| k})\) relating the jacobian ideal \(J_ X\) to the Kähler different \(\theta _ k(X/Y)\), where \(\pi: X\to Y\) is a separable noetherian normalization of X and \(\phi: \Omega ^ d_{X/k}\to \omega _ X\) is the canonical morphism from d-differentials to dualizing sheaf. Some corollaries are given, in particular: \((i)\quad J_ X\subset \tilde J_ X:=Ann(co\ker \phi),\) (ii) under suitable hypotheses the equality \(J_ X=\tilde J_ X\) implies that X is a local complete intersection. The proof of (*) is based on an algebraic result: let \(0\to N\to M\to M/N\to 0\) be an exact sequence of finitely generated modules over a ring S. Suppose N is generated by r elements. Then: \(F^ r(M) \bigwedge ^ rN=F^ 0(M/N) \bigwedge ^ rM\), where \(F^ i(-)\) denotes the i-th Fitting ideal of the S-module -.

MSC:

14K30 Picard schemes, higher Jacobians
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14H40 Jacobians, Prym varieties

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References:

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