##
**The ampleness of the theta divisor on the compactified Jacobian of proper and integral curve.**
*(English)*
Zbl 0824.14022

Let \(X\) be a proper and integral curve over an algebraically closed field \(k\). The moduli space of line bundles of degree zero on \(X\) is then, in a natural way, a group scheme over \(k\), called the generalized Jacobian \(P(X)\) of \(X\).

If \(X\) is a smooth curve, then \(P(X)\) of \(X\) is just the usual Jacobian of \(X\), i.e., an abelian variety. However, if \(X\) is singular, then \(P(X)\) is only a non-proper group scheme, and the question of whether there is a natural “compactification” of \(P(X)\), that is a proper scheme \(\overline P(X)\) containing \(P(X)\) as an open subset and on which \(P(X)\) acts as a group scheme, arises quite inevitably. In fact, such a (so-called) compactified Jacobian can be constructed, namely by considering the functor of families of torsion-free sheaves of rank one on \(X\). This idea of construction goes back to A. Grothendieck [cf. “Fondements de la géométrie algébrique,” Extraits du Séminaire Bourbaki 1957-1962 (Paris 1962; Zbl 0239.14002)]. Explicit constructions have been carried out, in the sequel, by A. Mayer and D. Mumford (1963), C. D’Souza (1973), T. Oda and C. S. Seshadri (1974), A. Altman, A. Iarrobino and S. Kleiman (1976), C. J. Rego (1980), and others. The fundamental paper of A. B. Altman and S. L. Kleiman “Compactifying the Picard scheme” [Adv. Math. 35, 50-112 (1980; Zbl 0427.14015)] gives the perhaps most general and complete account, with generalizations to higher dimensional varieties \(X\). Finally, the various attempts of constructing compactified Jacobians of curves have led to the now well-established theory of moduli spaces of semistable vector bundles over a curve.

The paper under review deals with the projectivity of the compactified Jacobian \(\overline P (X)\). Using the relative approach by A. Altman and S. Kleiman, the author constructs a Cartier divisor \(\Theta\) on \(\overline P^{g - 1} (X)\), the reduced \((g - 1)\)-component of the compactified Picard scheme \(\overline {\text{Pic}} (X/k)\) with respect to the arithmetic genus \(g\) of \(X\), and proves that \(\Theta\) is ample. – The proof is given in such a way that it can be generalized to the relative case, i.e., to morphisms \(f : X \to S\) of finite type, flat and projective, whose fibers are integral curves of arithmetic genus \(g\). In this situation, the author’s construction leads to a sheaf \({\mathcal D}\) on \(\overline {\text{Pic}}^{g - 1} (X/S)\) which is \(S\)-ample.

If \(X\) is a smooth curve, then \(P(X)\) of \(X\) is just the usual Jacobian of \(X\), i.e., an abelian variety. However, if \(X\) is singular, then \(P(X)\) is only a non-proper group scheme, and the question of whether there is a natural “compactification” of \(P(X)\), that is a proper scheme \(\overline P(X)\) containing \(P(X)\) as an open subset and on which \(P(X)\) acts as a group scheme, arises quite inevitably. In fact, such a (so-called) compactified Jacobian can be constructed, namely by considering the functor of families of torsion-free sheaves of rank one on \(X\). This idea of construction goes back to A. Grothendieck [cf. “Fondements de la géométrie algébrique,” Extraits du Séminaire Bourbaki 1957-1962 (Paris 1962; Zbl 0239.14002)]. Explicit constructions have been carried out, in the sequel, by A. Mayer and D. Mumford (1963), C. D’Souza (1973), T. Oda and C. S. Seshadri (1974), A. Altman, A. Iarrobino and S. Kleiman (1976), C. J. Rego (1980), and others. The fundamental paper of A. B. Altman and S. L. Kleiman “Compactifying the Picard scheme” [Adv. Math. 35, 50-112 (1980; Zbl 0427.14015)] gives the perhaps most general and complete account, with generalizations to higher dimensional varieties \(X\). Finally, the various attempts of constructing compactified Jacobians of curves have led to the now well-established theory of moduli spaces of semistable vector bundles over a curve.

The paper under review deals with the projectivity of the compactified Jacobian \(\overline P (X)\). Using the relative approach by A. Altman and S. Kleiman, the author constructs a Cartier divisor \(\Theta\) on \(\overline P^{g - 1} (X)\), the reduced \((g - 1)\)-component of the compactified Picard scheme \(\overline {\text{Pic}} (X/k)\) with respect to the arithmetic genus \(g\) of \(X\), and proves that \(\Theta\) is ample. – The proof is given in such a way that it can be generalized to the relative case, i.e., to morphisms \(f : X \to S\) of finite type, flat and projective, whose fibers are integral curves of arithmetic genus \(g\). In this situation, the author’s construction leads to a sheaf \({\mathcal D}\) on \(\overline {\text{Pic}}^{g - 1} (X/S)\) which is \(S\)-ample.

Reviewer: W.Kleinert (Berlin)

### MSC:

14H40 | Jacobians, Prym varieties |

14K30 | Picard schemes, higher Jacobians |

14D20 | Algebraic moduli problems, moduli of vector bundles |

### Keywords:

ampleness; polarization; generalized Jacobian; projectivity of the compactified Jacobian; compactified Picard scheme### References:

[1] | A. Altman , A. Iarrobino , S. Kleiman , Nordic Summer School/NAVF, Symposium in mathematics, Oslo 1976. |

[2] | A. Altman , S. Kleiman , Compactifying the Picard Scheme , Advances in Mutheniatics, Vol. 35, p. 50-112. · Zbl 0427.14015 |

[3] | S. Bosch , W. Lütkebohmert , M. Raynaud , Néron Models , Springer 1990. · Zbl 0705.14001 |

[4] | A. Grothendieck , Fondements de la Géométrie Algébrique, second exposé sur le foncteur de Picard . |

[5] | R. Hartshorne , Introduction to Algebraic Geometry , Springer 1978. · Zbl 0367.14001 |

[6] | D. Mumford , Geometric Invariant Theory , Springer 1965. · Zbl 0147.39304 |

[7] | C.J. Rego , The compactified Jacobian , Annales scientifiques E.N.S., t. 13 1980, p. 211-223. · Zbl 0478.14024 |

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