04.17.01 · algebraic-geometry / birational-geometry

Birational geometry and the minimal model program

shipped3 tiersLean: none

Anchor (Master): Kollár & Mori 1998 Birational Geometry of Algebraic Varieties (Cambridge Tracts 134); Matsuki 2002 Introduction to the Mori Program; Birkar–Cascini–Hacon–McKernan 2010 (J. Amer. Math. Soc. 23).

Intuition Beginner

Two shapes in algebraic geometry are birational when they become identical once you delete a small enough set from each. The function field — the field of rational functions — is the same object, and a small set of "bad" points does not change it. So birational equivalence is coarser than isomorphism: isomorphic shapes are birational, but birational shapes need not be isomorphic.

A blow-up is the basic move that relates two birational but non-isomorphic shapes. Pick a point on a surface and refuse to remember that distinct directions through it were collapsed; you replace the point with a whole copy of projective space — one point per direction. Nothing changes away from that point, so the function field is untouched, but the geometry upstairs is finer.

The minimal model program (MMP) asks the natural next question: in each birational class, what is the simplest representative? Repeatedly blow down anything that makes the shape more complicated than it has to be. For curves the answer is forced — every curve is already minimal. For surfaces Castelnuovo showed the process terminates at a minimal model. In higher dimensions the process can get stuck mid-step, and un-sticking it requires a flip.

Visual Beginner

Picture a flat sheet with a single marked point. The blow-up leaves every other point alone and replaces the marked point by a circle of directions — a copy of $ma t hbb P^{1}$ sitting where the point used to be. Two curves that crossed at the marked point now meet the circle at two different places; they have been separated.

The minimal model program runs this picture in reverse wherever possible: a circle with the right numerical signature ( $E^{2} = - 1$ , $E co n g ma t hbb P^{1}$ ) can be collapsed back to a smooth point, simplifying the surface.

Worked example Beginner

Blow up the affine plane $ma t hbb A^{2}$ at the origin. The result lives inside $ma t hbb A^{2} t im es ma t hbb P^{1}$ and is cut out by one equation, $x c d o t v = y c d o t u$ , where $[u : v]$ are coordinates on $ma t hbb P^{1}$ .

Above a point $(a, b) n e q (0, 0)$ the equation pins down $[u : v] = [a : b]$ , so exactly one point sits there — the blow-up is unchanged away from the origin. Above the origin the equation $0 = 0$ is automatic, and the entire $ma t hbb P^{1}$ survives. This copy of $ma t hbb P^{1}$ is the exceptional divisor $E$ .

The blow-up is smooth, and it is isomorphic to $ma t hbb A^{2}$ off the origin, so it is birational to $ma t hbb A^{2}$ without being isomorphic to it. The line $y = 2 x$ , which passed through the origin with slope $2$ , lifts to a curve meeting $E$ at the single direction $[1 : 2]$ . Two distinct lines through the origin, which met there downstairs, are now disjoint upstairs — each picks out its own direction.

The exceptional divisor $E co n g ma t hbb P^{1}$ has dimension $1$ . Replacing a point by a $ma t hbb P^{1}$ of directions is the local shape of every birational modification.

Check your understanding Beginner

Formal definition Intermediate+

Let $X, Y$ be integral varieties over an algebraically closed field $k$ .

Definition (rational and birational maps). A rational map $f : X d a s h r i g h t a r r o w Y$ is an equivalence class of pairs $(U, f_{U})$ where $U s u b se t e q X$ is a dense open and $f_{U} : U t o Y$ is a morphism, two pairs being identified when they agree on the overlap. The map $f$ is birational if there exists a dense open $U s u b se t e q X$ on which $f_{U}$ is an isomorphism onto a dense open of $Y$ . Equivalently, $f$ induces an isomorphism of function fields $k (Y) x r i g h t a r r o w s im k (X)$ .

Definition (birational equivalence). Two integral varieties $X, Y$ are birationally equivalent if there exists a birational map $X d a s h r i g h t a r r o w Y$ ; equivalently $k (X) co n g k (Y)$ .

A birational morphism $p i : w i d e t i l d e X t o X$ is a morphism that is birational as a rational map: an isomorphism over some dense open of $X$ . Blow-ups along closed subschemes are the prototypical birational morphisms ^{[Hartshorne §II.7]}, studied in detail in the blow-up unit 04.07.02. A blow-down is a left-inverse birational morphism contracting an exceptional divisor.

Definition (exceptional locus, discrepancy). Let $p i : w i d e t i l d e X t o X$ be a birational morphism of normal varieties. The exceptional locus $ma t h r m E x c (p i) s u b se t e q w i d e t i l d e X$ is the largest closed subset on which $p i$ fails to be a local isomorphism. For a prime divisor $E s u b se t e q w i d e t i l d e X$ and $X ma t hbb Q$ -Cartier, the discrepancy $a (E, X) in ma t hbb Q$ is defined by pulling back the canonical divisor:

K_{w i d e t i l d e X}; =; p i^{*} K_{X}; +; s u m_{E} a (E, X), E,

the sum ranging over $p i$ -exceptional prime divisors $E$ (the identity holds on a fixed common birational model).

Definition (singularities of the MMP). Let $X$ be a normal $ma t hbb Q$ -Gorenstein variety. Then $X$ is:

terminal if $a (E, X) > 0$ for every exceptional divisor $E$ over $X$ ;
canonical if $a (E, X) g e q 0$ for every exceptional divisor $E$ over $X$ ;
Kawamata log terminal (klt) if $a (E, X) > - 1$ for every divisor $E$ over $X$ and $l f l oor D e l t a r f l oor = 0$ for the relevant boundary (in the pair setting $(X, D e l t a)$ );
log canonical if $a (E, X) g e q - 1$ for every divisor $E$ over $X$ .

These thresholds form the singularity strata that the MMP is allowed to pass through. Terminal singularities are the mildest: a normal surface is terminal if and only if it is smooth.

Definition (canonical ring, Kodaira dimension). For a normal projective variety $X$ with canonical sheaf $o m e g a_{X}$ , the canonical ring is the graded $k$ -algebra

R (X, K_{X}); =; bi g o pl u s_{m g e q 0} H^{0}! l e f t (X,, o m e g a_{X}^{o t im es m} r i g h t) .

The Kodaira dimension $k a pp a (X) in - in f t y, 0, 1, l d o t s, d im X$ is

k a pp a (X); =; ma t h r m t r . d e g_{k}, ma t h r m F r a c, R (X, K_{X}); -; 1,

with $k a pp a (X) = - in f t y$ when $R (X, K_{X}) = k$ (no positive-degree pluri-forms). Equivalently $k a pp a (X)$ is the maximal dimension of the image of the rational maps $p h i_{∣ m K_{X} ∣} : X d a s h r i g h t a r r o w ma t hbb P^{P_{m} - 1}$ as $m$ varies, where $P_{m} = h^{0} (X, o m e g a_{X}^{o t im es m})$ is the $m$ -th plurigenus.

Definition (minimal model). A normal projective variety $X$ with terminal and canonical singularities is a minimal model if the canonical divisor $K_{X}$ is nef (numerically eventually finite: $K_{X} c d o tC g e q 0$ for every curve $C$ ). A Mori fibre space is a morphism $X t o S$ with relative Picard number one, $- K_{X}$ $p i$ -ample, and fibres Fano-type; the MMP dichotomy asserts every variety is birational either to a minimal model or to a Mori fibre space.

Key theorem with proof Intermediate+

Theorem (birational invariance of the plurigenera and Kodaira dimension). Let $p hi : X d a s h r i g h t a r r o w Y$ be a birational map between smooth projective varieties over $k$ . Then for every $m g e q 0$ the spaces of pluri-canonical forms are canonically isomorphic,

H^{0}! l e f t (Y,, o m e g a_{Y}^{o t im es m} r i g h t); co n g; H^{0}! l e f t (X,, o m e g a_{X}^{o t im es m} r i g h t) .

Consequently the canonical rings $R (X, K_{X})$ and $R (Y, K_{Y})$ are isomorphic as graded $k$ -algebras, and $k a pp a (X) = k a pp a (Y)$ .

The proof reduces, via resolution of indeterminacy, to the case of a birational morphism, where it becomes a push-down identity for the canonical sheaf.

Proof. Step 1: reduce to a birational morphism. By resolution of singularities and of indeterminacy ^{[Hironaka 1964]}, there exists a smooth projective variety $Z$ and birational morphisms $p : Z t o X$ , $q : Z t o Y$ resolving $p hi$ — i.e. $p hi c i r c p = q$ as rational maps. It is enough to show $H^{0} (Z, o m e g a_{Z}^{o t im es m}) co n g H^{0} (X, o m e g a_{X}^{o t im es m})$ for any birational morphism $p : Z t o X$ of smooth projective varieties; the same statement applied to $q$ then gives the chain of isomorphisms.

Step 2: the discrepancy decomposition. For a birational morphism $p : Z t o X$ of smooth varieties, the canonical divisors satisfy

K_{Z}; =; p^{*} K_{X}; +; s u m_{i} a_{i} E_{i},

where the $E_{i}$ are the $p$ -exceptional prime divisors and each coefficient $a_{i}$ is a strictly positive integer. The positivity $a_{i} > 0$ is the local computation of the Jacobian of $p$ near a point of $E_{i}$ : $p$ ramifies along $E_{i}$ , so $d e t (ma t h r m d p)$ vanishes to positive order there, and that order is $a_{i}$ . (For the blow-up of a smooth centre of codimension $c$ , $a_{i} = c - 1 g e q 1$ ; see the Full proof set.) Tensoring by $m g e q 1$ ,

o m e g a_{Z}^{o t im es m}; =; p^{*} o m e g a_{X}^{o t im es m}, o t im es, ma t h c a l O_{Z}! l e f t (t e x t s t y l e s u m_{i} m a_{i} E_{i} r i g h t),

with $s u m_{i} m a_{i} E_{i}$ an effective exceptional divisor.

Step 3: the push-down identity. We claim

p_{*} o m e g a_{Z}^{o t im es m}; =; o m e g a_{X}^{o t im es m} q q u a d (m g e q 1) .

By the projection formula, $p_{*} o m e g a_{Z}^{o t im es m} = o m e g a_{X}^{o t im es m} o t im es p_{*} ma t h c a l O_{Z} (s u m_{i} m a_{i} E_{i})$ , so it is enough to show $p_{*} ma t h c a l O_{Z} (D) = ma t h c a l O_{X}$ for every effective $p$ -exceptional $ma t hbb Q$ -divisor $D$ . Sections of $ma t h c a l O_{Z} (D)$ are regular functions on $Z se t min u s ma t h r m E x c (p)$ with poles of bounded order along $ma t h r m E x c (p)$ ; but $p$ identifies $Z se t min u s ma t h r m E x c (p) x r i g h t a r r o w s im X se t min u s p (ma t h r m E x c (p))$ , and $p (ma t h r m E x c (p))$ has codimension $g e q 2$ in $X$ (since each $E_{i}$ is a divisor contracted to lower dimension). Because $X$ is normal, regular functions on the complement of a codimension- $g e q 2$ subset extend (algebraic Hartogs), so such sections are regular on all of $X$ : $p_{*} ma t h c a l O_{Z} (D) = ma t h c a l O_{X}$ .

Step 4: conclude. Taking global sections,

H^{0} (Z, o m e g a_{Z}^{o t im es m}); =; H^{0} (X,, p_{*} o m e g a_{Z}^{o t im es m}); =; H^{0} (X, o m e g a_{X}^{o t im es m}),

an isomorphism natural in $X$ . Applying the same to $q$ gives $H^{0} (Y, o m e g a_{Y}^{o t im es m}) co n g H^{0} (Z, o m e g a_{Z}^{o t im es m}) co n g H^{0} (X, o m e g a_{X}^{o t im es m})$ , compatible with the graded ring structures. Hence $R (X, K_{X}) co n g R (Y, K_{Y})$ , and their fraction fields have the same transcendence degree, giving $k a pp a (X) = k a pp a (Y)$ . $s q u a r e$

Corollary. The Kodaira dimension $k a pp a$ is a birational invariant of smooth projective varieties, so it partitions the birational equivalence classes of varieties of each dimension into $d im X + 2$ strata.

Bridge. The birational invariance of the plurigenera builds toward the central organising principle of the entire strand: invariants of the function field — equivalently of the canonical ring — this is exactly the data the minimal model program seeks to recover. The same push-down identity $p_{*} o m e g a_{w i d e t i l d e X}^{o t im es m} = o m e g a_{X}^{o t im es m}$ appears again in the definition of terminal and canonical singularities, where the coefficients $a_{i}$ in $K_{w i d e t i l d e X} = p^{*} K_{X} + s u m a_{i} E_{i}$ become the discrepancy thresholds that organise the MMP. The construction generalises the genus of a curve to all dimensions, the central insight being that the canonical ring, not the variety, is the stable object, and the bridge is that every MMP step (blow-up, divisorial contraction, flip) preserves this ring while simplifying the underlying variety.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that the plurigenus $P_{m} (X) = h^{0} (X, o m e g a_{X}^{o t im es m})$ of a smooth projective curve $C$ of genus $g$ determines $g$ , and conclude from the birational invariance of $P_{m}$ that birational smooth projective curves are isomorphic.

Hint

For a curve, $d e g o m e g a_{C} = 2 g - 2$ . Use Riemann–Roch to compute $P_{m}$ for $m g e q 2$ .

Answer

For $m g e q 2$ , $d e g (o m e g a_{C}^{o t im es m}) = m (2 g - 2) > 0$ , so by Riemann–Roch $P_{m} (C) = h^{0} (o m e g a_{C}^{o t im es m}) = m (2 g - 2) - g + 1 = (2 m - 1) (g - 1)$ . Solving for $g$ : $g = 1 + P_{m} / (2 m - 1)$ . Thus $P_{m}$ determines $g$ for any $m g e q 2$ . By the birational-invariance theorem, birational smooth projective curves have equal $P_{m}$ and hence equal genus; and a birational map between smooth projective curves extends to an isomorphism (rational maps from smooth curves into projective varieties are morphisms), so they are isomorphic. In dimension one the birational class and the isomorphism class coincide.

Exercise 4 (medium, symbolic).

Let $p i : w i d e t i l d e X t o X$ be the blow-up of a smooth projective surface $X$ at a single point, with exceptional curve $E$ . Using $K_{w i d e t i l d e X} = p i^{*} K_{X} + E$ and $E^{2} = - 1$ , show that $K_{w i d e t i l d e X}^{2} = K_{X}^{2} - 1$ and $r h o (w i d e t i l d e X) = r h o (X) + 1$ , where $r h o$ is the Picard number.

Hint

Intersect $K_{w i d e t i l d e X}$ with itself, using $p i^{*} K_{X} c d o tE = 0$ (since $p i_{*} E = 0$ ) and $p i^{*} D_{1} c d o t p i^{*} D_{2} = D_{1} c d o t D_{2}$ .

Answer

Expand the self-intersection:

K_{w i d e t i l d e X}^{2} = (p i^{*} K_{X} + E)^{2} = (p i^{*} K_{X})^{2} + 2, p i^{*} K_{X} c d o tE + E^{2} = K_{X}^{2} + 0 + (- 1) = K_{X}^{2} - 1,

using $p i^{*} K_{X} c d o tE = 0$ (the pullback meets the contracted curve in zero) and the projection formula $(p i^{*} D)^{2} = p i^{*} (D^{2})$ pushed back, i.e. $= D^{2}$ .

For the Picard number: $ma t h r m P i c (w i d e t i l d e X) = p i^{*} ma t h r m P i c (X) o pl u s ma t hbb Z c d o t [E]$ (the exceptional class is not a pullback since it maps to a point), so $r h o (w i d e t i l d e X) = r h o (X) + 1$ . Each blow-up at a point raises the Picard number by one and lowers $K^{2}$ by one. This is why the minimal model of a surface — obtained by contracting all possible $(- 1)$ -curves — maximises $K^{2}$ and minimises $r h o$ within its birational class.

Exercise 6 (medium, symbolic).

Show that a birational map $ma t hbb P^{2} d a s h r i g h t a r r o w ma t hbb P^{1} t im es ma t hbb P^{1}$ is given by the pencil of conics through a fixed point, and explain why $ma t hbb P^{2}$ and $ma t hbb P^{1} t im es ma t hbb P^{1}$ are birational without being isomorphic.

Hint

Both have function field $k (s, t)$ . The conic pencil $l amb d a Q_{1} + m u Q_{2} = 0$ through a common point gives a rational map; away from the base locus it is a morphism.

Answer

Choose coordinates $[X : Y : Z]$ on $ma t hbb P^{2}$ with distinguished point $p = [0 : 0 : 1]$ . The pencil of lines through $p$ is $l amb d a X + m u Y = 0$ , parametrised by $[l amb d a : m u] in ma t hbb P^{1}$ . A line through $p$ meets a fixed line $e l l = Z = 0 co n g ma t hbb P^{1}$ in one further point, giving a morphism $ma t hbb P^{2} se t min u s p t o ma t hbb P^{1} t im es e l l co n g ma t hbb P^{1} t im es ma t hbb P^{1}$ sending $q$ to $(t e x t l in e t h r o ug h p, q;; q c a p e l l)$ . This is an isomorphism of dense opens, so $ma t hbb P^{2}$ and $ma t hbb P^{1} t im es ma t hbb P^{1}$ are birational: both have function field $k (s, t)$ . They are not isomorphic because $r h o (ma t hbb P^{2}) = 1$ while $r h o (ma t hbb P^{1} t im es ma t hbb P^{1}) = 2$ , and $ma t hbb P^{1} t im es ma t hbb P^{1}$ is the minimal model obtained from $ma t hbb P^{2}$ by blowing up one point and contracting the strict transform of the line through it — a elementary transformation (or link).

Exercise 7 (hard, symbolic).

Let $p i : w i d e t i l d e X t o X$ be a birational morphism of smooth projective surfaces. Prove that

K_{w i d e t i l d e X}; =; p i^{*} K_{X}; +; s u m_{i = 1}^{r} E_{i},

where $E_{1}, l d o t s, E_{r}$ are the exceptional curves (each $E_{i} co n g ma t hbb P^{1}$ , $E_{i}^{2} = - 1$ , and $E_{i} c d o t E_{j} = 0$ for $i n e q j$ ), and deduce that $p i$ factors as a sequence of point blow-ups.

Hint

By a theorem of Castelnuovo, every birational morphism of smooth surfaces factors into point blow-ups. First show the local discrepancy is $1$ at each exceptional curve by examining one blow-up.

Answer

Factorisation. By a structure theorem for birational surface morphisms (Castelnuovo, see Beauville Ch. II), every birational morphism $p i : w i d e t i l d e X t o X$ of smooth projective surfaces factors as

w i d e t i l d e X = X_{r} x r i g h t a r r o w p i_{r} X_{r - 1} t o c d o t s t o X_{1} x r i g h t a r r o w p i_{1} X_{0} = X,

with each $p i_{i}$ the blow-up of $X_{i - 1}$ at a single point $p_{i}$ , producing exceptional curve $E_{i}$ . This is the strong factorisation for surfaces, valid because surface singularities resolve by point blow-ups.

Discrepancy. Each point blow-up has discrepancy $1$ (Exercise 4 / codimension $c = 2$ , so $c - 1 = 1$ ). Adding the contributions along the chain, the total discrepancy on $w i d e t i l d e X$ over $X$ is the sum of the pullbacks of the individual $E_{i}$ , giving $K_{w i d e t i l d e X} = p i^{*} K_{X} + s u m_{i} E_{i}^{'}$ where $E_{i}^{'}$ is the total transform of the $i$ -th exceptional curve on $w i d e t i l d e X$ . In the basis of $w i d e t i l d e X$ the $E_{i}^{'}$ are precisely the final exceptional curves when the blow-up centres are general (no infinitely near points), each $E_{i} co n g ma t hbb P^{1}$ with $E_{i}^{2} = - 1$ and pairwise disjoint.

Consequence. The formula $K_{w i d e t i l d e X}^{2} = K_{X}^{2} - r$ holds, so the number of blow-ups is read off the canonical self-intersection: $r = K_{X}^{2} - K_{w i d e t i l d e X}^{2}$ . Combined with $r h o (w i d e t i l d e X) = r h o (X) + r$ , this pins down the birational geometry of a surface from two numerical invariants. $s q u a r e$

Exercise 8 (hard, conceptual).

State the cone theorem and the MMP dichotomy, and explain why a $K_{X}$ -negative small contraction (one that does not contract a divisor) cannot be the final step of the program and forces a flip.

Hint

A small contraction drops the Picard number relative to the canonical class but does not produce a $ma t hbb Q$ -Cartier canonical divisor on the target. A flip repairs this by a birational surgery that preserves the canonical ring.

Answer

Cone theorem (Mori). For a smooth projective variety $X$ , the Mori cone $o v er l in e ma t h r m N E (X)$ of effective curve classes is generated by $K_{X}$ -nef curves together with countably many $K_{X}$ -negative extremal rays $R_{i} = ma t hbb R_{g e q 0} [C_{i}]$ , each generated by a rational curve $C_{i}$ with $0 < - K_{X} c d o t C_{i} l e q 2 d im X$ . Each $K_{X}$ -negative extremal ray admits a contraction $ma t h r m C o n t_{R} : X t o Y$ .

The contraction is of three types: (i) Fano / Mori fibre space (the relative dimension jumps and $- K$ is relatively ample), (ii) divisorial (a divisor is contracted, the Picard number $r h o$ drops by $1$ , and $Y$ keeps canonical/terminal singularities), or (iii) small (no divisor is contracted; only curves are collapsed).

The MMP dichotomy: every smooth projective variety over a field of characteristic zero is birational either to a minimal model ( $K$ nef) or to a Mori fibre space ( $- K$ relatively ample).

Why a small contraction forces a flip. A small contraction $f : X t o Y$ collapses curves but no divisor, so the canonical divisor $K_{Y}$ fails to be $ma t hbb Q$ -Cartier (the push-forward of $K_{X}$ has no $ma t hbb Q$ -Cartier structure along the small image). The MMP cannot continue on $Y$ . A flip is a birational map $X d a s h r i g h t a r r o w X^{+}$ fitting into a commutative square with $Y$ on both sides, where $X^{+}$ has $ma t hbb Q$ -Cartier canonical class, $K_{X^{+}}$ is $f^{+}$ -ample, and the canonical rings agree: $R (X, K_{X}) co n g R (X^{+}, K_{X^{+}})$ . The flip replaces the $K_{X}$ -negative extremal ray by a $K_{X^{+}}$ -positive one, restoring the hypotheses needed to continue the program. Existence of flips was proved by Mori (dimension $3$ , 1988) and by Birkar–Cascini–Hacon–McKernan (log general type, all dimensions, 2010).

Lean formalization Intermediate+

lean_status: none — Mathlib contains the scheme-theoretic infrastructure for birational maps (function fields, rational maps, dense opens) and the blow-up construction via the Rees algebra, but the minimal model program is not formalised. The roadmap items — canonical divisor and ring, Kodaira dimension, discrepancies and the terminal/canonical/klt thresholds, the Mori cone, the cone theorem, and flips — are listed in Mathlib gap analysis.

-- Target statement (not yet formalisable in Mathlib without the imports
-- listed in lean_mathlib_gap):
--
-- theorem kodaira_dimension_birational_invariant
--   {k : Type*} [Field k] [IsAlgClosed k]
--   {X Y : Scheme} [IsSmoothProjectiveVarietyOver k X] [IsSmoothProjectiveVarietyOver k Y]
--   (φ : BirationalMap X Y) (m : ℕ) :
--     Module.equiv (H⁰ Y (Canonical.omega Y ^⊗ m)) (H⁰ X (Canonical.omega X ^⊗ m))
--
-- The key lemma is the push-down identity
--   π_* (Canonical.omega (Bl_Z X) ^⊗ m) ≅ Canonical.omega X ^⊗ m
-- for a birational morphism π : Bl_Z X → X of smooth projective varieties.

Advanced results Master

The Enriques–Kodaira classification of surfaces. For a smooth projective surface $X$ over $ma t hbb C$ , the Kodaira dimension takes values in $- in f t y, 0, 1, 2$ and rigidly organises the birational geometry ^{[Beauville 1996]}:

$k a pp a = - in f t y$ : $X$ is birationally ruled — birational to $ma t hbb P^{1} t im es C$ for some curve $C$ of genus $g$ . The minimal models are $ma t hbb P^{2}$ or the geometrically ruled surfaces $ma t hbb P (ma t h c a l E) t o C$ over a curve $C$ .
$k a pp a = 0$ : $K_{X}$ is numerically equivalent to zero in the minimal model. The four families of minimal surfaces with $k a pp a = 0$ are the K3 surfaces, the Enriques surfaces, the abelian surfaces, and the (hyperelliptic) bielliptic surfaces.
$k a pp a = 1$ : $X$ is properly elliptic — the pluri-canonical maps $p h i_{∣ m K ∣}$ give, for $m g g 0$ , an elliptic fibration $X t o ma t hbb P^{1}$ .
$k a pp a = 2$ : $X$ is of general type. The canonical ring $R (X, K_{X})$ is finitely generated, and $ma t h r m P r o j, R (X, K_{X})$ is the canonical model with ample canonical divisor and at most canonical singularities (rational double points).

The table is exhaustive in dimension two: every smooth projective surface lands in exactly one row, and the row is a birational invariant.

Higher-dimensional generalisation. In dimension $n$ the Kodaira dimension satisfies $- in f t y l e q k a pp a (X) l e q n$ . Varieties with $k a pp a = - in f t y$ are conjectured (and known in characteristic zero by BCHM combined with uniruledness theorems) to be uniruled — covered by rational curves. Varieties with $k a pp a = n$ are of general type; their canonical models exist and are the points of the coarse moduli of varieties of general type. The intermediate strata $0 l e q k a pp a < n$ correspond to a modular fibration $X t o B$ with $d im B = k a pp a$ and general fibre of $k a pp a = 0$ — the Iitaka fibration.

Terminal, canonical, and klt singularities. The discrepancy thresholds select the singularities the MMP must allow. Terminal singularities are the target of the MMP: in dimension two the only terminal surface singularities are smooth points; in dimension three the terminal singularities are isolated compound-Du Val points. Canonical singularities are slightly milder-allowing: rational double points in dimension two. Kawamata log terminal (klt) singularities are the natural setting for the log MMP with a boundary divisor $D e l t a$ : they are exactly the singularities for which vanishing theorems (Kawamata–Viehweg) and inversion of adjunction hold, and they are preserved under the MMP steps.

Flips and flops. A flip is the birational surgery required when a $K_{X}$ -negative extremal contraction is small (contracts no divisor); the flip diagram is

X x r i g h t a r r o w; f; Y x l e f t a r r o w; f^{+}; X^{+}, q q u a d K_{X^{+}} c d o t C^{+} > 0 t e x t o n t h e f l i pp e d r a y .

The existence of flips in dimension three was Mori's Fields-Medal theorem (1988) ^{[Mori 1988]}; the existence of log flips in all dimensions for varieties of log general type is the theorem of Birkar–Cascini–Hacon–McKernan (2010) ^{[BCHM 2010]}, which also proves finite generation of the canonical ring in general. A flop is the $K e q u i v 0$ analogue: a small birational modification $X d a s h r i g h t a r r o w X^{'}$ with $K_{X} e q u i v K_{X^{'}}$ numerically zero. Flops arise on Calabi–Yau varieties and more generally on varieties with $K e q u i v 0$ , and parametrise the birational flexible models within a fixed birational class satisfying $K e q u i v 0$ ; they are central to the minimal-model uniqueness question (two minimal models of a variety with $K e q u i v 0$ are connected by a sequence of flops, by Kawamata) and to derived equivalences (Bondal–Orlov predicts derived equivalence across a flop).

Synthesis. The minimal model program packages the entire birational classification problem into one inductive procedure on the canonical class. The central insight is that $K_{X}$ , once made numerically effective, is the correct generalisation of the genus of a curve to higher dimensions, and the discrepancy filtration $a (E, X)$ is exactly the device that selects which singularities the program may produce. Putting these together — the cone theorem (existence of $K$ -negative extremal rays), the contraction theorem, and the existence of flips (Mori in dimension three, BCHM in general) — one obtains the dichotomy: every variety is either birational to a minimal model ( $K$ nef) or to a Mori fibre space (a Fano-type fibration), and the foundational reason this terminates is that divisorial contractions drop the Picard number while flips are controlled by the ascending chain condition on the canonical ring. This is dual to the Enriques–Kodaira classification of surfaces, the bridge is that plurigenera are birational invariants, and the whole program generalises the Enriques–Kodaira table to arbitrary dimension; it appears again in the construction of moduli of varieties of general type (whose points are canonical models) and in the minimal model of the canonical ring itself.

Full proof set Master

Proposition (discrepancy of a blow-up along a smooth centre). Let $X$ be a smooth variety and $Z s u b se t e q X$ a smooth closed subvariety of codimension $c g e q 2$ . Let $p i : w i d e t i l d e X = ma t h r m B l_{Z} X t o X$ be the blow-up with exceptional divisor $E$ . Then $w i d e t i l d e X$ is smooth and

K_{w i d e t i l d e X}; =; p i^{*} K_{X}; +; (c - 1), E .

Proof. Work locally near a point of $Z$ . Choose algebraic coordinates $x_{1}, l d o t s, x_{c}, z_{1}, l d o t s, z_{n - c}$ on $X$ such that $Z$ is cut out by $x_{1} = c d o t s = x_{c} = 0$ ; the $z_{j}$ are coordinates along $Z$ and the $x_{i}$ are normal coordinates.

The blow-up is covered by $c$ affine charts; on the chart $U_{1}$ where the first homogeneous coordinate $u_{1} n e q 0$ , write $x_{i} = x_{1} c d o t v_{i}$ for $i = 2, l d o t s, c$ with $v_{i} = u_{i} / u_{1}$ . The coordinates on $U_{1}$ are $x_{1}, v_{2}, l d o t s, v_{c}, z_{1}, l d o t s, z_{n - c}$ , and the exceptional divisor is $E c a p U_{1} = x_{1} = 0$ .

A local generator of $o m e g a_{X}$ near the point is the volume form $O m e g a = d x_{1} w e d g e c d o t s w e d g e d x_{c} w e d g e d z_{1} w e d g e c d o t s w e d g e d z_{n - c}$ . Express this in the coordinates of $U_{1}$ : $d x_{i} = d (x_{1} v_{i}) = v_{i}, d x_{1} + x_{1}, d v_{i}$ for $i g e q 2$ , so

d x_{1} w e d g e d x_{2} w e d g e c d o t s w e d g e d x_{c} = d x_{1} w e d g e (v_{2} d x_{1} + x_{1} d v_{2}) w e d g e c d o t s w e d g e (v_{c} d x_{1} + x_{1} d v_{c}) .

Expanding, every term containing $d x_{1}$ twice vanishes; the surviving term is

d x_{1} w e d g e d x_{2} w e d g e c d o t s w e d g e d x_{c}; =; x_{1}^{c - 1}, d x_{1} w e d g e d v_{2} w e d g e c d o t s w e d g e d v_{c} .

Therefore the local generator of $o m e g a_{w i d e t i l d e X}$ on $U_{1}$ is

w i d e t i l d e O m e g a = d x_{1} w e d g e d v_{2} w e d g e c d o t s w e d g e d v_{c} w e d g e d z_{1} w e d g e c d o t s w e d g e d z_{n - c}; =; x_{1}^{- (c - 1)}, O m e g a .

Since $p i^{*} O m e g a = x_{1}^{c - 1} w i d e t i l d e O m e g a$ and $E c a p U_{1} = x_{1} = 0$ , the relation $w i d e t i l d e O m e g a = x_{1}^{- (c - 1)} O m e g a$ reads, as divisors,

K_{w i d e t i l d e X} bi g ∣_{U_{1}}; =; p i^{*} K_{X} bi g ∣_{U_{1}}; +; (c - 1) (E c a p U_{1}) .

The same computation on the other charts (with the roles of the $x_{i}$ rotated) gives the same coefficient $c - 1$ , so globally $K_{w i d e t i l d e X} = p i^{*} K_{X} + (c - 1) E$ . The coefficient $c - 1$ is the discrepancy $a (E, X)$ ; for a point blow-up on a surface $c = 2$ and $a (E, X) = 1$ , recovering the formula used in the exercises. $s q u a r e$

Corollary (surfaces are minimal up to $(- 1)$ -curves). On a smooth surface, every birational modification is a composition of point blow-ups (each with discrepancy $1$ ), so the only way to lower $K^{2}$ is to blow up a point, and the only way to raise it (within the smooth birational class) is to contract a $(- 1)$ -curve $E co n g ma t hbb P^{1}$ with $E^{2} = - 1$ . Castelnuovo's contraction theorem (1893) gives the converse: such a curve is the exceptional curve of a unique blow-down to a smooth surface. Iterating contractions of $(- 1)$ -curves terminates (since $K^{2}$ strictly increases and is bounded above), producing a minimal model of the surface — unique except when $X$ is rational or ruled, where two minimal models may differ by an elementary transformation.

Connections Master

Blow-up 04.07.02 — the blow-up $ma t h r m B l_{Z} X t o X$ is the elementary birational morphism and the local model for every MMP step. This unit reads off its discrepancy $(c - 1) E$ from the Jacobian computation that the blow-up unit sets up via the Rees algebra; the universal property of the blow-up is exactly the formal shadow of "the minimal surgery that turns an ideal into a Cartier divisor."
Hurwitz formula and the genus of curves 04.04.02 — in dimension one birational equivalence collapses to isomorphism, and the genus is the unique birational invariant. The Hurwitz formula $2 g_{Y} - 2 = n (2 g_{X} - 2) + d e g R$ is the one-dimensional shadow of the discrepancy decomposition $K_{w i d e t i l d e X} = p i^{*} K_{X} + s u m a_{i} E_{i}$ : the ramification divisor $R$ plays the role of the exceptional contribution, and both formulas are degree accounts of the same canonical pullback identity.
Scheme 04.02.01 — rational maps, birational morphisms, and the exceptional locus are all defined in the category of schemes, using function fields $k (X) = ma t h c a l O_{X, x i}$ at the generic point $x i$ . The MMP is a theorem about integral Noetherian schemes of finite type over a field, and discrepancies are computed on log resolutions obtained from blow-ups of schemes.
Riemann–Roch theorem for surfaces 04.05.08 — the Enriques–Kodaira classification is proved by combining the discrepancy formula $K_{w i d e t i l d e X} = p i^{*} K_{X} + E$ with the Riemann–Roch formula $c hi (ma t h c a l O_{X}) = t f r a c 112 (K_{X}^{2} + e (X))$ (Noether's formula) and the Hodge index theorem, pinning down the four strata from numerical invariants $K^{2}$ , $r h o$ , and $c hi (ma t h c a l O_{X})$ .
Hodge index theorem 04.05.09 — the signature of the intersection form on $ma t h r m P i c (X)$ of a surface (one positive direction, $r h o - 1$ negative) controls which curves can be contracted and underwrites the negativity of the exceptional locus, the input that lets Castelnuovo's contraction theorem identify exactly the $(- 1)$ -curves.
Sheaf cohomology 04.03.01 — the plurigenera $P_{m} (X) = h^{0} (X, o m e g a_{X}^{o t im es m})$ and the canonical ring are defined via coherent cohomology, and the birational-invariance proof turns on the push-down identity $p_{*} o m e g a_{w i d e t i l d e X}^{o t im es m} = o m e g a_{X}^{o t im es m}$ , a statement about higher direct images and the projection formula.

Historical & philosophical context Master

The classification of algebraic surfaces up to birational equivalence is the achievement of the Italian school — Castelnuovo, Enriques, and Severi — in the decade around 1900. Castelnuovo's contraction theorem (1893), identifying the $(- 1)$ -curves as exactly the curves a birational morphism of smooth surfaces may contract, is the operational core of the surface classification; Guido Castelnuovo and Federigo Enriques assembled the full birational classification table of surfaces over the following years. The Kodaira dimension was introduced by Kunihiko Kodaira in the 1960s in his analytic classification of compact complex surfaces ^{[Kodaira 1960s, Beauville 1996]}, and the resulting Enriques–Kodaira table unifies the algebraic and analytic pictures.

Oscar Zariski reformulated the Italian classification in the language of schemes in the 1940s and pushed the birational point of view into higher dimensions; he proved, in particular, that the birational classification of threefolds reduces to understanding terminal singularities and birational contractions. Heiskue Hironaka's 1964 theorem on resolution of singularities in characteristic zero ^{[Hironaka 1964]} supplied the tool the MMP depends on: any birational map can be resolved by a sequence of blow-ups along smooth centres, giving a common smooth model on which discrepancies are computed.

The modern minimal model program was created by Shigefumi Mori in the 1980s. Mori's cone theorem (1982) described the $K_{X}$ -negative extremal rays of the cone of curves, and his flip theorem (1988) proved the existence of flips for smooth threefolds ^{[Mori 1988]} — the result for which he received the Fields Medal in 1990. The general existence of flips and minimal models for varieties of log general type, in all dimensions, was established by Caucher Birkar, Alessio Cascini, Christopher Hacon, and James McKernan in 2010 ^{[BCHM 2010]}; Birkar and McKernan's later work extended parts of the program. The BCHM theorem simultaneously proves finite generation of the canonical ring, confirming a conjecture going back to Mori and Kawamata.

Philosophically the MMP reframes classification: instead of asking for an exhaustive list of varieties, one asks for a canonical representative of each birational class — the minimal model (or Mori fibre space) — obtained by an algorithmic sequence of canonical-class-driven surgeries. The canonical ring, not the variety, is the primary invariant; varieties of general type are classified by their canonical models, and the moduli of varieties of general type is the moduli of these canonical rings. The program remains open in positive characteristic in dimensions $g e q 4$ , where resolution of singularities itself is still conjectural.

Bibliography Master

@article{hironaka1964resolution,
  author  = {Hironaka, Heisuke},
  title   = {Resolution of singularities of an algebraic variety over a field of characteristic zero},
  journal = {Annals of Mathematics (2)},
  volume  = {79},
  year    = {1964},
  pages   = {109--326}
}

@article{mori1982threefolds,
  author  = {Mori, Shigefumi},
  title   = {Threefolds whose canonical bundles are not numerically effective},
  journal = {Annals of Mathematics (2)},
  volume  = {116},
  year    = {1982},
  pages   = {133--176}
}

@article{mori1988flip,
  author  = {Mori, Shigefumi},
  title   = {Flip theorem and the existence of minimal models for threefolds},
  journal = {Annals of Mathematics (2)},
  volume  = {127},
  year    = {1988},
  pages   = {281--328}
}

@article{bchm2010,
  author  = {Birkar, Caucher and Cascini, Alessio and Hacon, Christopher D. and McKernan, James},
  title   = {Existence of minimal models for varieties of log general type},
  journal = {Journal of the American Mathematical Society},
  volume  = {23},
  year    = {2010},
  pages   = {405--468}
}

@book{hartshorne1977,
  author    = {Hartshorne, Robin},
  title     = {Algebraic Geometry},
  series    = {Graduate Texts in Mathematics},
  number    = {52},
  publisher = {Springer},
  year      = {1977}
}

@book{kollar_mori1998,
  author    = {Koll{\'a}r, J{\'a}nos and Mori, Shigefumi},
  title     = {Birational Geometry of Algebraic Varieties},
  series    = {Cambridge Tracts in Mathematics},
  number    = {134},
  publisher = {Cambridge University Press},
  year      = {1998}
}

@book{matsuki2002,
  author    = {Matsuki, Kenji},
  title     = {Introduction to the Mori Program},
  publisher = {Springer},
  year      = {2002}
}

@book{beauville1996,
  author    = {Beauville, Arnaud},
  title     = {Complex Algebraic Surfaces},
  series    = {London Mathematical Society Student Texts},
  number    = {34},
  publisher = {Cambridge University Press},
  edition   = {2},
  year      = {1996}
}

@book{lazarsfeld2004,
  author    = {Lazarsfeld, Robert},
  title     = {Positivity in Algebraic Geometry {I}},
  publisher = {Springer},
  year      = {2004}
}

@article{zariski1940,
  author  = {Zariski, Oscar},
  title   = {Algebraic surfaces},
  journal = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  publisher = {Springer},
  year    = {1935; 2nd ed. 1971 with supplements by Lipman and D. Mumford}
}

@book{reid1988,
  author    = {Reid, Miles},
  title     = {Undergraduate Algebraic Geometry},
  series    = {London Mathematical Society Student Texts},
  number    = {12},
  publisher = {Cambridge University Press},
  year      = {1988}
}

Prerequisites

04.02.01
04.04.02

Tier anchors

beginner: Reid 1988 Undergraduate Algebraic Geometry (CUP) §4, informal: two varieties agree off a small set; Miles Reid's schoolroom account of blowing up.
intermediate: Hartshorne 1977 Algebraic Geometry (Springer GTM 52) §I.4–I.6 birational maps, §V surfaces and the Enriques classification; Beauville 1996 Complex Algebraic Surfaces (Cambridge) Ch. I–IV.
master: Kollár & Mori 1998 Birational Geometry of Algebraic Varieties (Cambridge Tracts 134); Matsuki 2002 Introduction to the Mori Program; Birkar–Cascini–Hacon–McKernan 2010 (J. Amer. Math. Soc. 23).

References

Hartshorne — Algebraic Geometry · Springer GTM 52 (1977). §I.4 rational and birational maps; §II.7 blowing up; §V.1–V.5 surfaces, the Enriques classification, and the intersection form on $Pic$ . Standard scheme-theoretic reference.
Kollár & Mori — Birational Geometry of Algebraic Varieties · Cambridge Tracts in Mathematics 134 (1998). Ch. 1–2 on birational morphisms, discrepancies, and terminal/canonical/klt singularities; Ch. 3 on the cone theorem, contractions, and flips.
Birkar, Cascini, Hacon, McKernan — Existence of minimal models for varieties of log general type · J. Amer. Math. Soc. 23 (2010), 405–468. The MMP existence theorem in all dimensions for pairs of log general type; the finite-generation theorem for the canonical ring.
Beauville — Complex Algebraic Surfaces · Cambridge LMS Student Texts 34 (1996). Ch. I–IV on birational classification of surfaces, the intersection form, Kodaira dimension, and the Enriques table.
Mori — Flip theorem and the existence of minimal models · Ann. of Math. (2) 127 (1988), 281–328. The cone theorem and the existence of flips in dimension three; foundational work of the modern MMP.
Hironaka — Resolution of singularities of an algebraic variety over a field of characteristic zero · Ann. of Math. (2) 79 (1964), 109–326. Resolution by blowups along smooth centres, the input that lets the MMP work with smooth models.
Lazarsfeld — Positivity in Algebraic Geometry I · Springer (2004). §2.2 on $Q$ -divisors, §2.3 on the canonical divisor, and the birational invariance of plurigenera.

Estimated time

beginner: 15m
intermediate: 45m
master: 90m