# Running a Minimal Model Program

Joaquín Moraga

Communicated by Notices Associate Editor Han-Bom Moon

The main aim of algebraic geometry is the understanding and classification of algebraic varieties: geometric objects that can be defined by polynomial equations. The Minimal Model Program (MMP) was initiated by the Italian school of algebraic geometers in the dawn of the last century. The MMP proposes a solution for the classification problem using birational geometry. Instead of classifying varieties per se, we first perform some surgeries on them: the so-called birational transformations. Although these surgeries change the object of study, they preseve the main characteristics and nature of the variety. Then, we try to find a canonical element in the birational class of our variety, i.e., an element that is better behaved than others in its class. Finally, we aim to study a canonical model on each birational class to develop a classification. In summary, the MMP aims to classify algebraic varieties up to birational transformations.

The MMP is a cornerstone development in the theory of higher-dimensional algebraic varieties. For this reason it has attracted hundreds of researchers around the world and it has made important connections with many other topics in mathematics: differential geometry, number theory, combinatorics, and topology.

By now the literature on the MMP is vast and sophisticated. The aim of this note is to introduce the reader to some basic concepts in algebraic geometry and explain some of the beautiful ideas of the MMP. Rather than explaining the technical details, we try to encompass the essence of the topic and the observations that lead to it.

## The Projective Space

The -dimensional projective space is the playground of algebraic geometers. This space parametrizes the set of lines through the origin in . We use coordinates

to represent points in the projective space. The coordinate 1 represents a nonzero point of a line in through the origin, so multiplying a nonzero parameter in all the entries of 1 do not change the point in . This means that the relation

holds for every . The projective space admits a description as a disjoint union:

where the first set is called the -dimensional affine space and is described by in the coordinates 1, while the set is called the hyperplane at infinity and is described by in the coordinates 1. The points that lie in the hyperplane at infinity are called points at infinity.

Similar to how the rails of a train track seem to intersect at the horizon, two parallel lines in will intersect at infinity when they are considered as lines in . This is the advantage that the projective space offers to us compared with the affine space. It allows us to describe phenomena that happen at infinity and so this phenomena cannot be described in the affine space.

The -dimensional projective space can be covered with subsets called affine charts:

Each of these charts is itself a -dimensional affine space, i.e., each is isomorphic to via

Thus, all points in the projective space, even the points that lie at infinity, are endowed with local affine coordinates that allows us to study the geometry around such a point.

## Smooth Projective Varieties

An affine variety is the set of points in where a finite set of polynomials vanishes. This means that there are polynomials such that

In other words, an affine variety is a subset of that can be described only using polynomial equations. For instance, the set is an affine variety in while is not an affine variety. The variety is also called the vanishing locus of the set of polynomials .

Similar to the case of affine spaces and projective spaces, it is natural to consider projective versions of affine varieties, i.e., varieties on which we can study points at infinity. To do so, we consider the vanishing locus in of polynomials in the variables . Evaluating polynomials on the projective space requires a more careful analysis. If a polynomial is zero at the point , then we must require that it is also zero when evaluating at every point for every nonzero. Indeed, the points and are the same in . Polynomials satisfying this property are known as homogeneous polynomials. A projective variety is a subset of that can be described using homogeneous polynomials in the variables . For instance, the Fermat curves:

are projective varieties in the projective plane .

A projective variety is said to be smooth if around every point the variety can be approximated with a linear subspace of of dimension . The number is known as the dimension of . Roughly speaking, the dimension of a smooth projective variety tells us in how many linearly independent directions we can move from a given point . In the case of a variety defined by a single polynomial equation , as in the case of the Fermat curves, this geometric condition is equivalent to asking that the partial derivatives of do not vanish simultaneously in .

The main aim of algebraic geometry is the understanding and classification of smooth projective varieties. As we explain in the following section, the smoothness condition allows us to define certain functions on that are convenient to encode the geometry of .

## Line Bundles

When studying a geometric object we can study it extrinsically by understanding what kind of functions can be defined on . Among these functions, line bundles play a special role. A line bundle on a smooth projective variety associates to each point a line:

that vary holomorphically with the given point . The union of all these lines can be put together to form an algebraic variety that is called the total space of the line bundle. The total space of the line bundle is endowed with a projection function that sends the line to the point . We often identify the line bundle with its total space . Every projective variety comes with its trivial line bundle in which there is no variation of the chosen line.

A global section of a line bundle is a holomorphic function for which . For instance, Liouville’s theorem implies that a global section of on a projective variety must be a constant function . However, other line bundles may carry very interesting space of global sections. The space of global sections of a line bundle is denoted by . A set of global sections determine a polynomial function

given by

The target of this polynomial function is a projective space rather than an affine space to avoid the ambiguity of choosing different isomorphisms in 3. We say that a line bundle is very ample if we can find sections such that the associated polynomial function is an embedding. Very ample line bundles are the most important line bundles on a variety as its sections can be used to reconstruct the variety.

The usual operations for vector spaces: dual, tensor, and wedge, can be generalized to the context of line bundles. Thus, given two line bundles and on a projective variety , we can tensor them to give rise to a new line bundle . The self-tensors of a line bundle are called powers of the line bundle and denoted .

## The Canonical Line Bundle

Can we canonically construct a line bundle for a given projective variety ? The answer is yes for smooth projective varieties.

Let be a -dimensional smooth projective variety. Because of the smoothness, for each point we can associate a tangent plane . These tangent spaces can be put together to construct the tangent bundle of the smooth projective variety . The cotangent bundle is defined to be the dual bundle of the tangent bundle, i.e., . The canonical line bundle of is the -th wedge of the cotangent bundle, i.e., . This construction only depends on the isomorphism class of and it does not depend on the embedding of into an ambient projective space . The name canonical line bundle is assigned as this construction does not depend on any choice, it is intrinsically defined from . Every smooth projective variety carries a canonical line bundle . Once we have constructed this line bundle, we may consider its powers . These line bundles are called the pluricanonical or anti-pluricanonical line bundles depending on the sign of the integer . If , then we recover the trivial line bundle . One of the most successful approaches to studying the geometry of is via the analysis of the global sections of the pluricanonical line bundles and the anti-pluricanonical line bundles.

## The Trichotomy

There are three basic classes of smooth projective varieties depending on the positivity of the canonical line bundle . Depending on which class they belong to, either , has many global sections for or has many global sections of . We say that a smooth projective variety is canonically polarized (resp. Fano) if is very ample for some (resp. ). We say that a smooth projective variety is Calabi–Yau if . For a Calabi–Yau variety the only global sections of the pluricanonical or anti-pluricanonical line bundle are constant sections.

Canonically polarized, Calabi–Yau, and Fano varieties are the three building blocks of all smooth algebraic varieties. They are the algebro-geometric versions of the notion of hyperbolic, parabolic, and elliptic geometry either from classical geometry or from differential geometry. Additionally, this analogy becomes a theorem via the theory of Kähler–Einstein metrics. They behave quite differently from almost any perspective: topological, geometrical, or arithmetic. For instance, the following theorem is due to Kobayashi (see Kob61).

On the other hand, Gromov proved that the fundamental group of a smooth Calabi–Yau variety is almost an abelian group (see Gro78).

It is not yet clear how to describe the fundamental group of smooth canonically polarized varieties.

## Smooth Projective Curves

A smooth projective curve is a -dimensional smooth projective variety. In this case, every smooth projective curve is either canonically polarized, Calabi–Yau, or Fano. A smooth Fano curve is isomorphic to the projective line that topologically is a -dimensional sphere. For instance, the Fermat curves and are both isomorphic to the projective line. Smooth Fano curves are also called rational curves. A smooth Calabi–Yau curve is isomorphic to a cubic hypersurface in , i.e., a smooth curve that is defined by a single homogeneous cubic polynomial in and . These curves are known as elliptic curves and topologically they are . For example, the Fermat curve is Calabi–Yau. For each , the Fermat curve is canonically polarized. Any smooth curve is homotopic to a Riemann surface. The genus of the curve is the number of handles of the Riemann surface. From the geometric perspective, we can define the genus of a smooth projective curve to be

i.e., the genus can be understood as the number of linearly independent global sections of the canonical line bundle. From this perspective, a smooth projective curve is Fano (resp. Calabi–Yau or canonically polarized) if and only if (resp. or ).

The degree-genus formula states that the genus of a smooth projective curve defined by a homogeneous polynomial of degree in equals

Thus, a Fermat curve is canonically polarized if and only if . The most topologically accurate way to draw the Fermat curve of degree is as a sphere with handles attached.

Before we keep discussing the classification problem, we review the concept of divisors.

## Divisors on Curves

Let be a line bundle over a smooth projective curve. A meromorphic section is a section that is holomorphic outside finitely many points of and has no essential singularities. In other words, the function takes a value on for each while at these special points it diverges to infinity. To this meromorphic section , we can associate a formal sum of points of as follows. Around each point that the meromorphic function takes value or it locally behaves as with or respectively. This value is called the order of at and is usually denoted by . Then to we can associate the finite formal sum of points

This finite formal sum of points in a curve is what we call a divisor on the curve . Each divisor on a smooth projective curve corresponds uniquely to a line bundle equipped with a meromorphic section up to rescaling factor. Thus, we can use finite formal combination of points to encode the information of line bundles and meromorphic sections. For instance, the divisor , i.e., the formal sum of all points with coefficient zero, corresponds to the trivial line bundle with the constant section.

The degree of a divisor is the sum of the coefficients that appear in the finite formal sum, i.e., . The degree of a line bundle on a curve is defined to be , where is a meromorphic section of and is defined via 4. This number is indeed independent of the meromorphic section. The degree of the canonical line bundle of a smooth projective curve is equal to . So, the sign of the degree of the canonical line bundle determines whether the curve is Fano, Calabi–Yau, or canonically polarized.

## Divisors on Smooth Projective Varieties

A divisor on a smooth projective variety is a finite formal combination , where the are integers and are subvarieties of codimension , i.e., they have dimension exactly one less than the variety . This definition generalizes the concept of divisors on curves to higher dimensions. The duality between divisors and line bundles with meromorphic functions indeed works on every smooth projective variety. The line bundle associated to a divisor is often denoted by . Two divisors that are associated to the same line bundle are said to be linearly equivalent. Linear equivalence forms an equivalence relation between divisors. Linearly equivalent divisors exhibit similar geometric properties. A canonical divisor on a variety , often denoted by , is a divisor that is associated with the canonical line bundle , i.e., . For instance, in the divisor is a canonical divisor, while the trivial divisor is a canonical divisor on an elliptic curve.

Let be a divisor on a smooth projective variety and be a curve. We may define the intersection number to be

This concept generalizes the naive counting of intersection points between and .

Two divisors and are said to be numerically equivalent if for every curve . Linearly equivalent divisors are numerically equivalent. The Néron–Severi group of a smooth projective variety is the group of divisors⁠Footnote1 modulo numerical equivalence. The following theorem is due to Severi.

1

The Picard rank, denoted by , is the rank of and it measures the dimension of the space of divisors on .

## Smooth Projective Surfaces

A smooth projective surface is a -dimensional smooth projective variety. The study of smooth projective surfaces is much more complicated than the study of smooth projective curves. The main reason for this difficulty is that smooth projective surfaces admit certain surgeries that can change the isomorphism class of the surface but leave a dense open subset unchanged. This construction is known as blow-up and we proceed to explain it below.

We can consider the smooth surface

and let

The variety is isomorphic to . The surface admits a projection function . This projection function induces an isomorphism on . Further, the function restricted to is constant and maps the whole curve to the origin. The curve is called the exceptional curve of the blow-up. The morphism is what we call the blow-up of at the origin. In the previous example, the variety is usually denoted by . Note that the divisor represents the tangent directions of the origin in . In a few words, a blow-up is a surgery on a variety that cuts a subvariety and replaces it with a variety that represents the tangent directions of inside .

A priori, the blow-up construction poses a problem for the classification of smooth projective surfaces. Given any smooth projective surface , for instance the projective space , we may take any finite sequence of points and blow-up the points consecutively. By doing so, we obtain a sequence of surfaces with projection functions that are blow-ups. Each surface is not isomorphic to the previous ones, but there is an open dense subset which is isomorphic to an open dense subset of . In this case, we say that is birational to . A morphism from to that induces an isomorphism on open subsets is called a birational morphism.

Even for the study of the isomorphism classes of smooth projective varieties that are birational to it is a complicated task.

## Minimal Surfaces

The previous analysis hints toward the intuitive approach: instead of classifying surfaces up to isomorphism, we aim to study smooth projective surfaces that are not blow-ups of other smooth projective surfaces. This leads to the concept of minimal surface. A minimal surface is, roughly speaking, a surface that is not the blow-up of another smooth projective surface.

Castelnuovo proved the following theorem that characterizes minimal surfaces.

In the previous theorem the self-intersection is defined as . Smooth rational curves with are known as -curves. Castelnuovo theorem tells us that a surface is minimal if and only if it does not contain -curves. Furthermore, if a smooth projective surface contains a -curve, then such a curve can be blown-down to a new smooth projective surface . By doing so, the Picard rank, which is a positive integer, drops by one. This means that . Thus one cannot blow-down infinitely many -curves; this process is guaranteed to stop with a minimal surface after a finite number of steps. In summary, the Castelnuovo theorem is a very useful technique for the birational classification of surfaces: it postulates that any smooth projective surface can be transformed into a minimal smooth projective surface via blow-downs. Hence, for any smooth projective surface there is a birational morphism to a minimal smooth projective surface such that the preimages of are either points or connected union of smooth rational curves. This approach reduces the birational classification of smooth projective surfaces to the classification of minimal smooth projective surfaces. Minimal smooth projective surfaces were classified in the 1950’s by Kodaira and Enriques.

## Cone of Curves

A -cycle on a smooth projective variety is a finite formal combination of curves , where the are real numbers. We can define the intersection of a divisor with a -cycle by linearity:

We say that two -cycles and are numerically equivalent, written , if for every divisor . In other words, numerically equivalent -cycles are combinations of curves that cannot be distinguished by intersecting with divisors.

The space of curves, denoted by is the space of -cycles modulo numerical equivalence. This -vector space turns out to be the dual of the space of divisors defined as . The elements of are often called -line bundles. The intersection product

induces a perfect pairing.

A -cycle is said to be effective if each is nonnegative. The cone of curves of a smooth projective variety, denoted by , is the cone inside spanned by all the numerical classes containing an effective -cycle. The cone of curves defines a positive direction in the space of curves. For instance, for , we have

where is the class of a straight line in .

If we blow-up points on a surface, then the space of curves increases in dimension and the situation gets more interesting. The space of curves of is -dimensional and is spanned by two curves: the exceptional curve and any curve whose image on is a straight line passing through . The blow-up of in at most randomly chosen points is a Fano variety and its cone of curves is polyhedral. On the contrary, the cone of curves of the blow-up of at general points is not polyhedral.

## Cone Theorem

An element in a cone is called extremal if whenever with , then , and span the same ray in . A curve in a smooth projective variety is said to be an extremal curve if it is extremal in the closed cone of curves denoted by .⁠Footnote2 For example, -curves on surfaces are extremal.

2

When talking about extremal curves it is more natural to take the closure of the cone. An open cone does not have extremal rays.

For a smooth projective variety we can naturally split the closed cone of curves into three pieces: the -negative curves, the -trivial curves, and the -positive curves. The cone theorem states that the -negative region of the cone of curves carries a pleasant structure (see KM98).

In the previous theorem, we write for the set of elements with .

In the case of a smooth Fano variety the cone theorem implies that the closed cone of curves is a polyhedral cone. This is not true for Calabi–Yau varieties or canonically polarized varieties. For instance, if is a general elliptic curve, then is a smooth Calabi–Yau surface whose closed cone of curves is not polyhedral. Let be the diagonal curve, let be a closed point, and define and . Then, the curves , and generate, over , the space of curves. Hence, any -cycle is numerically equivalent to for some real parameters , and . In these coordinates, the cone of curves is described by

Thus, the closed cone of curves is a circular cone. In this case, by the definition of Calabi–Yau varieties, every -cycle intersects trivially.

## Contraction Theorem

In the case of a smooth projective surface every -curve satisfies that . Furthermore, every -curve is extremal. Hence, every -curve appears in the right-most summand of equality 5. Castelnuovo’s theorem then says that this curve can be contracted to a point to obtain a new smooth projective surface. The following theorem, known as the contraction theorem, states that a similar surgery can be performed on higher-dimensional smooth projective varieties (see KM98).

Geometrically, the contraction theorem asserts that all the curves that belong to an extremal -negative ray can be collapsed to points via a polynomial function. The adjective projective in the previous statement roughly means that the preimages are unions of projective varieties. The morphism , often called a contraction morphism, may be denoted by , where is a curve contained in the ray .

Let and be two distinct points in and let be the blow-up of at these two points. Let and be the preimages of and in the blow-up, respectively. Both and are surfaces isomorphic to . The closed cone of curves is generated by three curves: a straight line in , a straight line in , and the unique line on whose image in is the unique line through and . The canonical divisor satisfies

Theorem 6 can be applied to find two projective morphisms and . A variation of Theorem 6 can be applied to find a contraction that is an isomorphism in and contracts to a singular point of locally given by the equation : the so-called rational double point.

## The Three Types of Contractions

There are three types of contraction morphisms depending on the dimension of the set swept out by the curves with .

We say that is a Mori fiber space if . In this case, the curves that belong to the ray sweep out the whole variety , i.e., they cover the whole variety. If is a randomly chosen point of the base, then is a smooth projective Fano variety. In other words, a Mori fiber space is a way to cover the variety using Fano varieties in such a manner that the Fano varieties are compatible with the preimages of a polynomial function. Simple examples of Mori fiber spaces are projections , where is a Fano variety of Picard rank one. A projective bundle over a smooth projective variety is also an example of a Mori fiber space. A Mori fiber space is a generalization of the concept of projective bundle where the projective space is replaced with a Fano variety.

We say that is a divisorial contraction if and the locus where is not an isomorphism, the so-called exceptional locus, has dimension . In other words, the set swept out by the curves is a divisor of . A divisorial contraction is simply the opposite of a blow-up. In this case, we have that and while can be singular, the singularities of are not bad singularities. For instance, the divisor still induces a -line bundle so intersection theory with is well defined. Furthermore, both the cone theorem and contraction theorem still hold for .

We say that is a flipping contraction if and the exceptional locus has dimension at most . Flipping contractions only exist in dimension at least . In this case, the singularities of are bad singularities, for instance does not induce a -line bundle. Thus, the cone theorem and contraction theorem do not make sense on . In this case, we aim to replace with a different variety. Flipping contractions have a close relative: flopping contractions. The definition is the same but we require . For instance, the contraction described above is a flopping contraction.

## Flops

Let be the blow-up of at two points and . Let be the unique straight line in whose image on is the unique line through and . Using the notation introduced above, we may find a flopping contraction that only contracts the curve . The image of in is a singular point, the so-called rational double point locally given by the equation