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.

1. By Hironaka's resolution theorem, for each surface $S$, there is a birational projective morphism $S^{\prime }\to S$ from a smooth projective surface $S^{\prime }$.
2. By Castnonval's theorem, one can blow-down $(-1)$-curves on $S^{\prime }$, and get another smooth surface.
3. There are two results:
   1. Mori fibre spaces;
   2. $K_S$ 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
  1. $f$ is an isomorphism over $X\setminus (\mathrm{Sing}(X)\cup \mathrm{Supp}Z)$;
  2. $f^{\ast }{\mathcal{I}}_Z\subset {\mathcal{O}}_Y$ is an invertible sheaf ${\mathcal{O}}_Y(-D)$;
  3. $\mathrm{Ex}(f)\cup D$ 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 $\sum _i{d}_i{D}_i$ is a $\mathbb{Z}$-linear combination of prime divisors $D_i$. It is effective if $d_i\ge 0$
• ${\mathcal{O}}_X(D)=\{f\in K(X):\mathrm{div}(f)+D\ge 0\}$
• 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

canonical divisor

Pairs

Resolutions

singularities

Main theorems

X-method

Flip and others