Math 2.0

Algebraic geometry Definition

Minimal model program

Builds on Projective varietyCanonical bundleNef line bundleBirational mapCone theoremCastelnuovo's contraction theorem

Definition

Let XX be a normal projective variety over a field of characteristic 00 with ℚ\Q-factorial terminal singularities. Let KXK_X be its canonical divisor, the divisor class of the canonical bundle. The variety XX is a minimal model if KXK_X is nef. The minimal model program applied to XX is the following procedure.

  1. If KXK_X is nef, stop: XX is a minimal model.
  2. Otherwise, by the cone theorem there is an extremal ray RR of the closed cone of curves of XX with KX⋅R<0K_X \cdot R < 0. It has a contraction φR ⁣:X→Z\varphi_R \colon X \to Z of exactly the curves whose classes lie in RR.
  3. If dim⁡Z<dim⁡X\dim Z < \dim X, stop: φR\varphi_R is a Mori fibre space.
  4. If φR\varphi_R is birational and its exceptional locus is a divisor, replace XX by ZZ and go back to step 1.
  5. If φR\varphi_R is birational and its exceptional locus has codimension at least 22, replace XX by the flip X+X^+ of φR\varphi_R. Then go back to step 1.

Remarks

In steps 4 and 5 the new variety again has ℚ\Q-factorial terminal singularities, which is why the program allows these singularities. The program terminates when there is no infinite sequence of flips. Flips exist in all dimensions by a theorem of Birkar, Cascini, Hacon and McKernan, and termination of flips is known in low dimensions, including dimension 33, but open in general. Example: for a smooth projective surface, there are no small contractions, and step 4 contracts a (−1)(-1)-curve by Castelnuovo's contraction theorem. The surface ℙ2\mathbb{P}^2 is not a minimal model, since Kℙ2⋅ℓ=−3<0K_{\mathbb{P}^2} \cdot \ell = -3 < 0 for a line ℓ\ell. The program stops at step 3 for it, with ZZ a point, so ℙ2\mathbb{P}^2 is a Mori fibre space. A smooth curve of genus g≥1g \ge 1 is a minimal model, since deg⁡K=2g−2≥0\deg K = 2g - 2 \ge 0.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

Show the path as a graph
Algebraic surface, concept, no evidence levelAlgebraic surfaceconceptAlgebraically closed field, concept, no evidence levelAlgebraically closed …conceptAmple line bundle, concept, no evidence levelAmple line bundleconceptBirational map, concept, no evidence levelBirational mapconceptBlowing up, concept, no evidence levelBlowing upconceptCanonical bundle, concept, no evidence levelCanonical bundleconceptIntersection theory, concept, no evidence levelIntersection theoryconceptNef line bundle, concept, no evidence levelNef line bundleconceptProjective space, concept, no evidence levelProjective spaceconceptProjective variety, concept, no evidence levelProjective varietyconceptCastelnuovo's contraction theorem, statement, no evidence levelCastelnuovo's contrac…statementCone theorem, statement, no evidence levelCone theoremstatementMinimal model program, definition, no evidence levelMinimal model programdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic surface concept
  2. Algebraically closed field concept
  3. Ample line bundle concept
  4. Birational map concept
  5. Blowing up concept
  6. Canonical bundle concept
  7. Intersection theory concept
  8. Nef line bundle concept
  9. Projective space concept
  10. Projective variety concept
  11. Castelnuovo's contraction theorem statement
  12. Cone theorem statement
  13. Minimal model program definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Minimal_model_program
About this page
Tag32VAKG7CPS
URIm2://32VAKG7CPS
Kinddefinition
WritingW0 stub
MSC14E30
DefinesMinimal model program
UsesProjective variety, Canonical bundle, Nef line bundle, Birational map, Cone theorem, Castelnuovo's contraction theorem

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.