MMP¶
In this class, we consider normal varities over \(\mathbb{C}\).
Intro¶
Our goal is to classify algebraic varieties up to birational equivalence, then describe their moduli if exists.
For \(\dim 1\) case, i.e. for integral curves, clearly we can choose the normalization of the curve to be the primitive in the birational class, which is a smooth projective varity. Then we construct the moduli space \(M_g\) of smooth curves of genus \(g\).
The motivation of MMP comes from \(\dim 2\) case, that is normal (quasi-)projective surfaces.
- By Hironaka's resolution theorem, for each surface \(S\), there is a birational projective morphism \(S'\to S\) from a smooth projective surface \(S'\).
- By Castnonval's theorem, one can blow-down \((-1)\)-curves on \(S'\), and get another smooth surface.
- There are two results:
- Mori fibre spaces;
- \(K_S\) nef, called minimal surface.
In higher dimensional cases, we have Cone Theorem and Contraction Theorem, therefore we can do the similar contractions.
- Let \(X\) be an irreducible reduced algebraic variety and \(\mathcal{I} \subset \mathcal{O}_X\) a coherent sheaf of ideals defining a closed subscheme \(Z\). Then there is a smooth variety \(Y\) and a projective morphism \(f: Y\to X\) such that
- \(f\) is an isomorphism over \(X \setminus (\mathrm{Sing}\,(X) \cup \mathrm{Supp}\,Z)\);
- \(f^*\mathcal{I}_Z \subset \mathcal{O}_Y\) is an invertible sheaf \(\mathcal{O}_Y(-D)\);
- \(\mathrm{Ex}\,(f) \cup D\) is an snc divisor.
- Cone theorem and Contraction theorem
However, there are difficulties:
- For a contraction \(f:X\to Y\), even \(X\) is smooth, \(Y\) may not be smooth but with singularities;
- Small contraction;
- Termination of the program.
Solution to the difficulties:
- Consider terminal varities. Furthermore, lc pairs;
- filps
- only partially solved: MMP with scaling for certain pairs (BCHM)
pairs¶
Divisors¶
- A prime Weil divisor or simply a prime divisor \(D \subset X\) on \(X\) is an integral subscheme in \(X\) with codimension \(1\).
- A Weil divisor \(\sum_id_iD_i\) is a \(\mathbb{Z}\)-linear combination of prime divisors \(D_i\). It is effective if \(d_i \geq 0\)
- \(\mathcal{O}_X(D)=\{f\in K(X): \mathrm{div}(f)+D\geq 0\}\)
- A Cartier is a Weil divisor with invertible ideal sheaf.
Two generalization:
1. \(\mathbb{R}\)-coefficients
2. relative version
Intersection¶
KM98 prop 1.36 1.35; JK 1.4.3 etc
Definition of \(\overline{\mathrm{NE}}(X)\)
Properties of Divisors¶
cohomology
TODO: definitions of ample, nef, big;
TODO: criterion for ampleness etc
TODO: kodaira lemma etc
Cone of divisors¶
cones of divisors
relative case