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Abundance Conjecture

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In algebraic geometry, the abundance conjecture is a conjecture of birational geometry. It predicts that if the canonical bundle   of a projective variety   is positive in an appropriate sense, then   has an "abundance" of sections.

Statement of the conjecture

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The simplest form of the conjecture is as follows.

Abundance conjecture. Let   be a smooth projective variety. If   is nef, then it is semi-ample: that is, the line bundle   is globally generated, or equivalently the linear system   is basepoint-free, for some  .

In other words, if   is nef, then its section are "abundant" enough to determine a morphism of   to projective space.

Log abundance

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The conjecture also has a "logarithmic" version, as is common in birational geometry.

Log abundance conjecture. Let   be a projective variety. Suppose   is an effective divisor on   such that the pair   is log canonical. If   is nef, then it is semi-ample.

Here log canonical is one of the classes of singularites of pairs encountered in minimal model theory. Roughly speaking, it is conjecturally the largest class of singularities which is closed under the operations of the log minimal model program.

History and status

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Surfaces.

For three-dimensional varieties in characteristic 0, the conjecture was proved by Miyaoka and Kawamata. The logarithmic version was proved by Keel, Matsuki, and McKernan.

References

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  • Kawamata, Yujiro (1992). "Abundance theorem for minimal threefolds". Invent. Math. 108 (2): 229–246. doi:10.1007/BF02100604.
  • Keel, Séan (1993). "Log abundance theorem for threefolds". Duke Math. J. 75 (1): 99–119. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  • Kollár, Janos (1998). Birational geometry of algebraic varieties. Cambridge University Press. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)




Iitaka conjecture

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To go to Iitaka conjecture.

In algebraic geometry, the Iitaka conjecture is an important conjecture of birational geometry which attempts to describe the relationship between the Kodaira dimensions of the base, fibre, and total space in an algebraic fibre space.

Statement of the conjecture

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Let f: XZ be an algebraic fibre space with X and Z smooth projective varieties, and let F be a general fibre of f. Then κ(X) ≥ κ(Z)+κ(F), where κ denotes the Kodaira dimension.

History and status

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  • Iitaka
  • Kawamata


Gorenstein and Cohen–Macaulay schemes

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Explain cohomological significance: Gorenstein => dualising line bundle, Cohen–Macaulay => dualising sheaf