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Threefolds whose canonical bundles are not numerically effective.

S Mori1

  • 1Mathematics Department, Harvard University, Cambridge, Massachusetts 02138.

Proceedings of the National Academy of Sciences of the United States of America
|June 1, 1980
PubMed
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Nonsingular projective 3-folds lacking a numerically effective canonical bundle are classified. These 3-folds either contain exceptional divisors, map to surfaces with conic fibers, or map to curves with Del Pezzo surface fibers, or are Fano 3-folds.

Area of Science:

  • Algebraic Geometry
  • Complex Geometry
  • Enumerative Geometry

Background:

  • The classification of algebraic varieties based on the properties of their canonical bundle is a central theme in modern algebraic geometry.
  • Understanding the structure of 3-folds with specific properties, such as a non-numerically effective canonical bundle, is crucial for developing a complete picture of the landscape of 3-folds.

Purpose of the Study:

  • To classify nonsingular projective 3-folds whose canonical bundle is not numerically effective.
  • To provide an explicit classification of the possible structures these 3-folds can possess.

Main Methods:

  • The study employs techniques from birational geometry and the theory of minimal models.
  • Classification relies on analyzing the behavior of the canonical bundle and its associated maps.

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Main Results:

  • The research establishes that such 3-folds fall into four distinct categories.
  • These categories include the presence of exceptional divisors, specific types of morphisms to surfaces or curves, or the 3-fold being a Fano variety.

Conclusions:

  • The classification provides a comprehensive understanding of nonsingular projective 3-folds with a non-numerically effective canonical bundle.
  • The results contribute to the broader program of classifying higher-dimensional algebraic varieties.