Math 2.0

Algebraic geometry Birational geometry Theorem

Cone theorem

Builds on Projective varietyCanonical bundleIntersection theoryAmple line bundleAlgebraically closed field

Statement

Let kk be an algebraically closed field and let XX be a smooth projective variety over kk of dimension nn. Let KXK_X be its canonical divisor (see canonical bundle). Let N1(X)N_1(X) be the real vector space of 11-cycles on XX modulo numerical equivalence. Let NE‾(X)⊆N1(X)\overline{NE}(X) \subseteq N_1(X) be the closure of the cone spanned by the classes of curves. Write NE‾(X)KX≥0={z∈NE‾(X):KX⋅z≥0}\overline{NE}(X)_{K_X \ge 0} = \set{z \in \overline{NE}(X) : K_X \cdot z \ge 0}, using the intersection pairing. Then there are countably many rational curves Ci⊂XC_i \subset X with 0<−KX⋅Ci≤n+10 < -K_X \cdot C_i \le n + 1 such that NE‾(X)=NE‾(X)KX≥0+∑iℝ≥0[Ci].\overline{NE}(X) = \overline{NE}(X)_{K_X \ge 0} + \sum_i \R_{\ge 0} [C_i]. Moreover, for every ample divisor HH and every ε>0\varepsilon > 0, only finitely many of the CiC_i are needed to write NE‾(X)=NE‾(X)KX+εH≥0+∑i finiteℝ≥0[Ci].\overline{NE}(X) = \overline{NE}(X)_{K_X + \varepsilon H \ge 0} + \sum_{i \text{ finite}} \R_{\ge 0} [C_i].

Remarks

The theorem is due to Mori. The rational curves come from his bend-and-break method: a curve on which KXK_X is negative deforms a lot and breaks off rational curves. The method works first in characteristic pp and then passes to characteristic 00. Example: on ℙn\mathbb{P}^n the space N1N_1 is spanned by the class of a line ℓ\ell, and −K⋅ℓ=n+1-K \cdot \ell = n + 1. So the bound n+1n + 1 is sharp, and NE‾(ℙn)=ℝ≥0[ℓ]\overline{NE}(\mathbb{P}^n) = \R_{\ge 0} [\ell]. In characteristic 00, the theorem extends to log terminal pairs, together with the contraction theorem which contracts each KXK_X-negative extremal ray.

Going deeper

For a student

The cone theorem is the first step of the minimal model program: it provides the extremal rays to contract. With the contraction theorem, each KXK_X-negative extremal ray gives a morphism that contracts exactly the curves in that ray. For surfaces, these rays are spanned by (−1)(-1)-curves, by fibres of a ruling, or by a line in ℙ2\mathbb{P}^2.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Algebraically closed field, concept, no evidence levelAlgebraically closed …conceptAmple line bundle, concept, no evidence levelAmple line bundleconceptCanonical bundle, concept, no evidence levelCanonical bundleconceptIntersection theory, concept, no evidence levelIntersection theoryconceptProjective variety, concept, no evidence levelProjective varietyconceptCone theorem, statement, no evidence levelCone theoremstatement
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.
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  1. Algebraically closed field concept
  2. Ample line bundle concept
  3. Canonical bundle concept
  4. Intersection theory concept
  5. Projective variety concept
  6. Cone theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cone_of_curves
About this page
TagZJSW0HP75H
URIm2://ZJSW0HP75H
Kindstatement, theorem
WritingW0 stub
MSC14E30, 14C20
StatusThis copy of the site has no build data. Run m2 status ZJSW0HP75H.
AboutCanonical bundle
UsesProjective variety, Canonical bundle, Intersection theory, Ample line bundle, Algebraically closed field

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