Running a Minimal Model Program

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 $n$-dimensional projective space $\mathbb{P}^n$ is the playground of algebraic geometers. This space parametrizes the set of lines through the origin in $\mathbb{C}^{n+1}$. We use coordinates

$$\begin{equation} [x_0:\dots :x_n] \in \mathbb{P}^n {\tag{1}} \end{equation}$$

to represent points in the projective space. The coordinate 1 represents a nonzero point of a line in $\mathbb{C}^{n+1}$ through the origin, so multiplying a nonzero parameter $\lambda$ in all the entries of 1 do not change the point in $\mathbb{P}^n$. This means that the relation

$$\begin{equation} [x_0:\dots :x_n]=[\lambda x_0:\dots :\lambda x_n] {\tag{2}} \end{equation}$$

holds for every $\lambda \in \mathbb{C}^*$. The projective space admits a description as a disjoint union:

$$\begin{equation*} \mathbb{P}^n = \mathbb{C}^n \bigsqcup \mathbb{P}^{n-1}, \end{equation*}$$

where the first set $\mathbb{C}^n$ is called the $n$-dimensional affine space and is described by $x_n\neq 0$ in the coordinates 1, while the set $\mathbb{P}^{n-1}_\mathbb{C}$ is called the hyperplane at infinity and is described by $x_n=0$ 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 $\mathbb{C}^2$ will intersect at infinity when they are considered as lines in $\mathbb{P}^2_\mathbb{C}$. 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 $n$-dimensional projective space $\mathbb{P}^n$ can be covered with $n+1$ subsets called affine charts:

$$\begin{equation*} H_i\coloneq \{ [x_0:\dots :x_n]\mid x_i \neq 0\}. \end{equation*}$$

Each of these charts is itself a $n$-dimensional affine space, i.e., each $H_i$ is isomorphic to $\mathbb{C}^n$ via

$$\begin{equation*} [x_0:\dots :x_n]\mapsto (x_j/x_i)_{j\neq i}. \end{equation*}$$

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 $X$ is the set of points in $\mathbb{C}^n$ where a finite set of polynomials vanishes. This means that there are polynomials $p_1(x_1,\dots ,x_n),\dots ,p_k(x_1,\dots ,x_n)$ such that

$$\begin{equation*} X=\{ (x_1,\dots ,x_n) \mid p_i(x_1,\dots ,x_n)=0 \text{ for } 1\leq i\leq k \}. \end{equation*}$$

In other words, an affine variety is a subset of $\mathbb{C}^n$ that can be described only using polynomial equations. For instance, the set $\{ (x,y) \mid x^2+y^3=0\}$ is an affine variety in $\mathbb{C}^2$ while $\{ (x,y) \mid x=\sin y\}$ is not an affine variety. The variety $X$ is also called the vanishing locus of the set of polynomials $p_1,\dots ,p_k$.

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 $\mathbb{P}^n$ of polynomials $p_i$ in the variables $x_0,\dots ,x_n$. Evaluating polynomials on the projective space requires a more careful analysis. If a polynomial $p$ is zero at the point $(x_0,\dots ,x_n)$, then we must require that it is also zero when evaluating at every point $(\lambda x_0,\dots ,\lambda x_n)$ for every $\lambda$ nonzero. Indeed, the points $[x_0:\dots :x_n]$ and $[\lambda x_0:\dots : \lambda x_n]$ are the same in $\mathbb{P}^n$. Polynomials satisfying this property are known as homogeneous polynomials. A projective variety is a subset of $\mathbb{P}^n$ that can be described using homogeneous polynomials in the variables $x_0,\dots ,x_n$. For instance, the Fermat curves:

$$\begin{equation*} C_k\coloneq \{[x_0:x_1:x_2] \mid x_0^k+x_1^k+x_2^k=0\} \end{equation*}$$

are projective varieties in the projective plane $\mathbb{P}^2$.

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

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 $X$ that are convenient to encode the geometry of $X$.

Line Bundles

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

$$\begin{equation} \mathcal{L}_x\simeq \mathbb{C} {\tag{3}} \end{equation}$$

that vary holomorphically with the given point $x\in X$. The union of all these lines $\mathcal{L}_x$ can be put together to form an algebraic variety $\mathcal{L}$ that is called the total space of the line bundle. The total space of the line bundle is endowed with a projection function $p_{\mathcal{L}} \colon \mathcal{L}\rightarrow X$ that sends the line $\mathcal{L}_x$ to the point $x$. We often identify the line bundle with its total space $\mathcal{L}$. Every projective variety $X$ comes with its trivial line bundle $\mathcal{O}_X$ in which there is no variation of the chosen line.

A global section of a line bundle $\mathcal{L}$ is a holomorphic function $s\colon X\rightarrow \mathcal{L}$ for which $p_\mathcal{L}\circ s=\mathrm{id}_X$. For instance, Liouville’s theorem implies that a global section of $\mathcal{O}_X$ on a projective variety $X$ must be a constant function $X\rightarrow \mathbb{C}$. However, other line bundles may carry very interesting space of global sections. The space of global sections of a line bundle is denoted by $\Gamma (\mathcal{L})$. A set of global sections $s_0,\dots ,s_n\in \Gamma (\mathcal{L})$ determine a polynomial function

$$\begin{equation*} \phi \colon X \mathrel{\rightdasharrow }\mathbb{P}^n, \end{equation*}$$

given by

$$\begin{equation*} x\mapsto [s_0(x):\dots :s_n(x)]. \end{equation*}$$

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 $\mathcal{L}$ is very ample if we can find sections $s_0,\dots ,s_n \in \Gamma (\mathcal{L})$ such that the associated polynomial function $\phi$ 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 $\mathcal{L}$ and $\mathcal{M}$ on a projective variety $X$, we can tensor them to give rise to a new line bundle $\mathcal{L}\otimes \mathcal{M}$. The self-tensors of a line bundle $\mathcal{L}$ are called powers of the line bundle and denoted $\mathcal{L}^m$.

The Canonical Line Bundle

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

Let $X\subset \mathbb{P}^n$ be a $d$-dimensional smooth projective variety. Because of the smoothness, for each point $x\in X$ we can associate a tangent plane $T_{X,x}$. These tangent spaces can be put together to construct the tangent bundle $T_X$ of the smooth projective variety $X$. The cotangent bundle $\Omega _X$ is defined to be the dual bundle of the tangent bundle, i.e., $\Omega _X\coloneq T_X^\vee$. The canonical line bundle of $X$ is the $n$-th wedge of the cotangent bundle, i.e., $\omega _X\coloneq \wedge ^n \Omega _X$. This construction only depends on the isomorphism class of $X$ and it does not depend on the embedding of $X$ into an ambient projective space $\mathbb{P}^n$. The name canonical line bundle is assigned as this construction does not depend on any choice, it is intrinsically defined from $X$. Every smooth projective variety $X$ carries a canonical line bundle $\omega _X$. Once we have constructed this line bundle, we may consider its powers $\omega _X^m$. These line bundles are called the pluricanonical or anti-pluricanonical line bundles depending on the sign of the integer $m$. If $m=0$, then we recover the trivial line bundle $\mathcal{O}_X$. One of the most successful approaches to studying the geometry of $X$ 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 $\omega _X$. Depending on which class they belong to, either $\omega _X\simeq \mathcal{O}_X$, $\omega _X^m$ has many global sections for $m\gg 0$ or $\omega _X^m$ has many global sections of $m\ll 0$. We say that a smooth projective variety is canonically polarized (resp. Fano) if $\omega _X^m$ is very ample for some $m>0$ (resp. $m<0$). We say that a smooth projective variety is Calabi–Yau if $\omega _X\simeq \mathcal{O}_X$. 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).

Theorem 1.

Let $X$ be a smooth Fano variety and let $x\in X$ be a point. Then $\pi _1(X;x)\simeq \{1\}$.

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

Theorem 2.

Let $X$ be a smooth Calabi–Yau variety of dimension $n$ and $x\in X$ be a point. Then $\pi _1(X;x)$ admits a normal abelian subgroup of rank at most $2n$ and finite index.

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 $1$-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 $\mathbb{P}^1$ that topologically is a $2$-dimensional sphere. For instance, the Fermat curves $C_1$ and $C_2$ 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 $\mathbb{P}^2$, i.e., a smooth curve that is defined by a single homogeneous cubic polynomial in $x_0,x_1,$ and $x_2$. These curves are known as elliptic curves and topologically they are $S^1\times S^1$. For example, the Fermat curve $C_3$ is Calabi–Yau. For each $k\geq 4$, the Fermat curve $C_k$ is canonically polarized. Any smooth curve $C$ is homotopic to a Riemann surface. The genus $g(C)$ 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 $C$ to be

$$\begin{equation*} g(C)\coloneq \dim _\mathbb{C} \Gamma (\omega _C), \end{equation*}$$

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 $C$ is Fano (resp. Calabi–Yau or canonically polarized) if and only if $g(C)=0$ (resp. $g(C)=1$ or $g(C)\geq 2$).

The degree-genus formula states that the genus of a smooth projective curve $C$ defined by a homogeneous polynomial of degree $d$ in $\mathbb{P}^2$ equals

$$\begin{equation*} g(C)=\frac{(d-1)(d-2)}{2}. \end{equation*}$$

Thus, a Fermat curve $C_d$ is canonically polarized if and only if $d\geq 4$. The most topologically accurate way to draw the Fermat curve of degree $4$ is as a sphere with $3$ handles attached.

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

Divisors on Curves

Let $\mathcal{L}\rightarrow C$ be a line bundle over a smooth projective curve. A meromorphic section $s\colon C\mathrel{\rightdasharrow }\mathcal{L}$ is a section that is holomorphic outside finitely many points of $C$ and has no essential singularities. In other words, the function takes a value on $\mathcal{L}_c\simeq \mathbb{C}$ for each $c\in C$ while at these special points it diverges to infinity. To this meromorphic section $s$, we can associate a formal sum of points of $C$ as follows. Around each point $c\in C$ that the meromorphic function takes value $0$ or $\infty$ it locally behaves as $z\mapsto z^m$ with $m>0$ or $m<0$ respectively. This value $m$ is called the order of $s$ at $p\in C$ and is usually denoted by $m_p$. Then to $s$ we can associate the finite formal sum of points

$$\begin{equation} D(s)\coloneq \sum _{p\in C}m_pP. {\tag{4}} \end{equation}$$

This finite formal sum of points in a curve is what we call a divisor on the curve $C$. Each divisor on a smooth projective curve $C$ corresponds uniquely to a line bundle $\mathcal{L}$ equipped with a meromorphic section $s$ 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 $0$, i.e., the formal sum of all points with coefficient zero, corresponds to the trivial line bundle $\mathcal{O}_X$ with the constant section.

The degree of a divisor $D=\sum _{p\in C}m_p P$ is the sum of the coefficients that appear in the finite formal sum, i.e., $\operatorname {deg}(D)=\sum _{p\in C}m_p$. The degree of a line bundle $\mathcal{L}$ on a curve $C$ is defined to be $\deg (\mathcal{L})\coloneq \deg (D(s))$, where $s$ is a meromorphic section of $\mathcal{L}$ and $D(s)$ is defined via 4. This number is indeed independent of the meromorphic section. The degree of the canonical line bundle $\omega _C$ of a smooth projective curve $C$ is equal to $2g-2$. 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 $D$ on a smooth projective variety $X$ is a finite formal combination $\sum _{V\subset X} m_V V$, where the $m_V$ are integers and $V\subset X$ are subvarieties of codimension $1$, i.e., they have dimension exactly one less than the variety $X$. 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 $D$ is often denoted by $\mathcal{O}_X(D)$. 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 $X$, often denoted by $K_X$, is a divisor that is associated with the canonical line bundle $\omega _X$, i.e., $\omega _X\simeq \mathcal{O}_X(K_X)$. For instance, in $\mathbb{P}^1$ the divisor $-\{0\}-\{\infty \}$ is a canonical divisor, while the trivial divisor is a canonical divisor on an elliptic curve.

Let $D$ be a divisor on a smooth projective variety $X$ and $C\subset X$ be a curve. We may define the intersection number to be

$$\begin{equation*} D\cdot C\coloneq \deg (\mathcal{O}_X(D)|_C). \end{equation*}$$

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

Two divisors $D$ and $D'$ are said to be numerically equivalent if $(D-D')\cdot C=0$ for every curve $C\subset X$. Linearly equivalent divisors are numerically equivalent. The Néron–Severi group $\mathrm{NS}(X)$ of a smooth projective variety $X$ is the group of divisors⁠¹ modulo numerical equivalence. The following theorem is due to Severi.

¹

The group structure being addition.

✖

Theorem 3.

The Néron–Severi group $\mathrm{NS}(X)$ of a smooth projective variety is a finitely generated abelian group.

The Picard rank, denoted by $\rho (X)$, is the rank of $\mathrm{NS}(X)$ and it measures the dimension of the space of divisors on $X$.

Smooth Projective Surfaces

A smooth projective surface is a $2$-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

$$\begin{equation*} X\coloneq \{((x,y),[z:w])\mid xw-zy=0\} \subset \mathbb{C}^2 \times \mathbb{P}^1, \end{equation*}$$

and let

$$\begin{equation*} E\coloneq \{((0,0),[z:w])\}\subset X. \end{equation*}$$

The variety $E$ is isomorphic to $\mathbb{P}^1$. The surface $X$ admits a projection function $\pi \colon X\rightarrow \mathbb{C}^2$. This projection function induces an isomorphism on $X\setminus E \rightarrow \mathbb{C}^2\setminus \{(0,0)\}$. Further, the function $\pi$ restricted to $E$ is constant and maps the whole curve to the origin. The curve $E$ is called the exceptional curve of the blow-up. The morphism $\pi$ is what we call the blow-up of $\mathbb{C}^2$ at the origin. In the previous example, the variety $X$ is usually denoted by $\operatorname {Bl}_{(0,0)}\mathbb{C}^2$. Note that the divisor $E\simeq \mathbb{P}^1$ represents the tangent directions of the origin in $\mathbb{C}^2$. In a few words, a blow-up is a surgery on a variety $X$ that cuts a subvariety $Z$ and replaces it with a variety $E$ that represents the tangent directions of $Z$ inside $X$.

A priori, the blow-up construction poses a problem for the classification of smooth projective surfaces. Given any smooth projective surface $X$, for instance the projective space $\mathbb{P}^2$, we may take any finite sequence of points $p_1,\dots ,p_k\in \mathbb{P}^2$ and blow-up the points consecutively. By doing so, we obtain a sequence of surfaces $X_0\coloneq X,X_1,X_2,\dots ,X_k$ with projection functions $X_i \rightarrow X_{i-1}$ that are blow-ups. Each surface $X_i$ is not isomorphic to the previous ones, but there is an open dense subset $U_i\subset X_i$ which is isomorphic to an open dense subset of $X$. In this case, we say that $X_i$ is birational to $X$. A morphism from $X_i$ to $X$ that induces an isomorphism on open subsets is called a birational morphism.

Even for $\mathbb{P}^2$ 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.

Theorem 4.

Let $X$ be a smooth projective surface. Let $E\subset X$ be a smooth rational curve with $E^2=-1$. Then, there exists a projection morphism $\pi \colon X \rightarrow Y$ to a smooth projective surface $Y$ that induces an isomorphism $X\setminus E\simeq Y\setminus \pi (E)$ and $\pi (E)$ is a point $y\in Y$.

In the previous theorem the self-intersection $E^2$ is defined as $\mathcal{O}_X(E)\cdot E$. Smooth rational curves $E$ with $E^2=-1$ are known as $(-1)$-curves. Castelnuovo theorem tells us that a surface is minimal if and only if it does not contain $(-1)$-curves. Furthermore, if a smooth projective surface $X$ contains a $(-1)$-curve, then such a curve can be blown-down to a new smooth projective surface $Y$. By doing so, the Picard rank, which is a positive integer, drops by one. This means that $\rho (Y)=\rho (X)-1$. Thus one cannot blow-down infinitely many $(-1)$-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 $X$ there is a birational morphism $\pi \colon X\rightarrow X_{\text{min}}$ to a minimal smooth projective surface $X_{\text{min}}$ such that the preimages of $\pi$ 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 $1$-cycle on a smooth projective variety $X$ is a finite formal combination of curves $C_0\coloneq \sum _{C\subset X} m_C C$, where the $m_c$ are real numbers. We can define the intersection of a divisor $D$ with a $1$-cycle $C_0$ by linearity:

$$\begin{equation*} D\cdot C_0 = \sum _{C\subset X} m_C(D\cdot C). \end{equation*}$$

We say that two $1$-cycles $C_0$ and $C_1$ are numerically equivalent, written $C_0\equiv C_1$, if $D\cdot (C_0-C_1)=0$ for every divisor $D$. In other words, numerically equivalent $1$-cycles are combinations of curves that cannot be distinguished by intersecting with divisors.

The space of curves, denoted by $N_1(X)$ is the space of $1$-cycles modulo numerical equivalence. This $\mathbb{R}$-vector space turns out to be the dual of the space of divisors defined as $N^1(X)\coloneq \mathrm{NS}(X)\otimes _\mathbb{Z} \mathbb{R}$. The elements of $N^1(X)$ are often called $\mathbb{R}$-line bundles. The intersection product

$$\begin{equation*} N^1(X)\times N_1(X)\rightarrow \mathbb{R} \qquad (D,C) \mapsto D\cdot C \end{equation*}$$

induces a perfect pairing.

A $1$-cycle $C_0=\sum _{C\subset X} m_c C$ is said to be effective if each $m_c$ is nonnegative. The cone of curves of a smooth projective variety, denoted by $\operatorname {NE}(X)$, is the cone inside $N_1(X)$ spanned by all the numerical classes containing an effective $1$-cycle. The cone of curves defines a positive direction in the space of curves. For instance, for $n\geq 2$, we have

$$\begin{equation*} N_1(\mathbb{P}^n)\simeq \mathbb{R} \text{ and } \operatorname {NE}(\mathbb{P}^n) \simeq \mathbb{R}_{\geq 0}[\ell ] \end{equation*}$$

where $[\ell ]$ is the class of a straight line $\ell$ in $\mathbb{P}^n$.

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 $\operatorname {Bl}_p(\mathbb{P}^2)$ is $2$-dimensional and $\operatorname {NE}(\operatorname {Bl}_p(\mathbb{P}^2))$ is spanned by two curves: the exceptional curve $E$ and any curve $H$ whose image on $\mathbb{P}^2$ is a straight line passing through $p$. The blow-up of $\mathbb{P}^2$ in at most $8$ 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 $\mathbb{P}^2$ at $9$ general points is not polyhedral.

Cone Theorem

An element $v$ in a cone $\sigma$ is called extremal if whenever $v=v_0+v_1$ with $v_0,v_1\in \sigma$, then $v,v_0$, and $v_1$ span the same ray in $\sigma$. A curve $C$ in a smooth projective variety $X$ is said to be an extremal curve if it is extremal in the closed cone of curves denoted by $\overline{\operatorname {NE}}(X)$.⁠² For example, $(-1)$-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 $X$ we can naturally split the closed cone of curves $\overline{\operatorname {NE}}(X)$ into three pieces: the $K_X$-negative curves, the $K_X$-trivial curves, and the $K_X$-positive curves. The cone theorem states that the $K_X$-negative region of the cone of curves carries a pleasant structure (see KM98).

Theorem 5.

Let $X$ be an $n$-dimensional smooth projective variety. Then, there are countably many rational curves $C_i$, with $0<-K_X\cdot C_i\leq n+1$ such that

$$\begin{equation} \overline{\operatorname {NE}}(X)=\overline{\operatorname {NE}}(X)_{K_X\geq 0}+\sum _i \mathbb{R}_{\geq 0}[C_i]. {\tag{5}} \end{equation}$$

Furthermore, the rays $\mathbb{R}_{\geq 0}[C_i]$ only accumulate to the hyperplane $K_X^{\perp }$.

In the previous theorem, we write $\overline{\operatorname {NE}}(X)_{K_X\geq 0}$ for the set of elements $[C]\in \overline{\operatorname {NE}}(X)$ with $K_X\cdot C \geq 0$.

In the case of a smooth Fano variety $X$ the cone theorem implies that the closed cone of curves $\overline{\operatorname {NE}}(X)$ is a polyhedral cone. This is not true for Calabi–Yau varieties or canonically polarized varieties. For instance, if $E$ is a general elliptic curve, then $E\times E$ is a smooth Calabi–Yau surface whose closed cone of curves is not polyhedral. Let $\delta \subset E\times E$ be the diagonal curve, let $p\in E$ be a closed point, and define $f_1\coloneq [p\times E]$ and $f_2\coloneq [E\times p]$. Then, the curves $f_1,f_2$, and $\delta$ generate, over $\mathbb{R}$, the space of curves. Hence, any $1$-cycle $C$ is numerically equivalent to $xf_1+yf_2+z\delta$ for some real parameters $x,y$, and $z$. In these coordinates, the cone of curves is described by

$$\begin{equation*} \{ (x,y,z) \mid x+y+z\geq 0 \text{ and } xy+xz+yz\geq 0 \}\subset N_1(X). \end{equation*}$$

Thus, the closed cone of curves is a circular cone. In this case, by the definition of Calabi–Yau varieties, every $1$-cycle intersects $K_X$ trivially.

Contraction Theorem

In the case of a smooth projective surface every $(-1)$-curve $E$ satisfies that $K_X\cdot E=-1$. Furthermore, every $(-1)$-curve is extremal. Hence, every $(-1)$-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).

Theorem 6.

Let $X$ be a smooth projective variety. Let $R \subset \overline{\operatorname {NE}}(X)$ be an extremal ray that is $K_X$-negative. Then, there exists a projective morphism $\phi _R \colon X\rightarrow Y$ with the following property: the image of a curve $C\subset X$ on $Y$ is a point if and only if $[C]\in R$.

Geometrically, the contraction theorem asserts that all the curves that belong to an extremal $K_X$-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 $\phi _R$, often called a contraction morphism, may be denoted by $\phi _C$, where $C$ is a curve contained in the ray $R$.

Let $p$ and $q$ be two distinct points in $\mathbb{P}^3$ and let $X=\operatorname {Bl}_{p,q}(\mathbb{P}^3)$ be the blow-up of $\mathbb{P}^3$ at these two points. Let $E_p$ and $E_q$ be the preimages of $p$ and $q$ in the blow-up, respectively. Both $E_p$ and $E_q$ are surfaces isomorphic to $\mathbb{P}^2$. The closed cone of curves $\overline{\operatorname {NE}}(X)$ is generated by three curves: a straight line $\ell _p$ in $E_p$, a straight line $\ell _q$ in $E_q$, and the unique line $\ell _0$ on $X$ whose image in $\mathbb{P}^3$ is the unique line through $p$ and $q$. The canonical divisor $K_X$ satisfies

$$\begin{equation*} K_X\cdot \ell _p = -2, \quad K_X\cdot \ell _q = -2, \text{ and } K_X\cdot \ell _0 = 0. \end{equation*}$$

Theorem 6 can be applied to find two projective morphisms $\phi _{\ell _p}\colon X\rightarrow \mathrm{Bl}_q(\mathbb{P}^3)$ and $\phi _{\ell q}\colon X\rightarrow \operatorname {Bl}_p(\mathbb{P}^3)$. A variation of Theorem 6 can be applied to find a contraction $\phi _{\ell _0}\colon X\rightarrow X_0$ that is an isomorphism in $X\setminus \ell _0$ and contracts $\ell _0$ to a singular point of $X_0$ locally given by the equation $xy-zw=0$: the so-called rational double point.

The Three Types of Contractions

There are three types of contraction morphisms $\phi _R$ depending on the dimension of the set swept out by the curves $C$ with $[C]\in R$.

We say that $\phi _R \colon X \rightarrow Y$ is a Mori fiber space if $\dim Y < \dim X$. In this case, the curves that belong to the ray $R$ sweep out the whole variety $X$, i.e., they cover the whole variety. If $y\in Y$ is a randomly chosen point of the base, then $F=\phi _R^{-1}(y)$ is a smooth projective Fano variety. In other words, a Mori fiber space is a way to cover the variety $X$ 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 $X\times F \rightarrow X$, where $F$ 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 $\phi _R\colon X \rightarrow Y$ is a divisorial contraction if $\dim Y=\dim X$ and the locus $E$ where $\phi _R$ is not an isomorphism, the so-called exceptional locus, has dimension $\dim E= \dim X-1$. In other words, the set swept out by the curves $[C]\in R$ is a divisor of $X$. A divisorial contraction is simply the opposite of a blow-up. In this case, we have that $\rho (Y)=\rho (X)-1$ and while $Y$ can be singular, the singularities of $Y$ are not bad singularities. For instance, the divisor $K_Y$ still induces a $\mathbb{Q}$-line bundle so intersection theory with $K_Y$ is well defined. Furthermore, both the cone theorem and contraction theorem still hold for $Y$.

We say that $\phi _R\colon X\rightarrow Y$ is a flipping contraction if $\dim Y=\dim X$ and the exceptional locus $E$ has dimension at most $\dim X-2$. Flipping contractions only exist in dimension at least $3$. In this case, the singularities of $Y$ are bad singularities, for instance $K_Y$ does not induce a $\mathbb{Q}$-line bundle. Thus, the cone theorem and contraction theorem do not make sense on $Y$. In this case, we aim to replace $Y$ with a different variety. Flipping contractions have a close relative: flopping contractions. The definition is the same but we require $K_X\cdot R=0$. For instance, the contraction $\phi _{\ell _0}\colon X \rightarrow X_0$ described above is a flopping contraction.

Flops

Let $X$ be the blow-up of $\mathbb{P}^3$ at two points $p$ and $q$. Let $\ell _0$ be the unique straight line in $X$ whose image on $\mathbb{P}^3$ is the unique line through $p$ and $q$. Using the notation introduced above, we may find a flopping contraction $\phi _{\ell _0}\colon X \rightarrow X_0$ that only contracts the curve $\ell _0$. The image of $\ell _0$ in $X_0$ is a singular point, the so-called rational double point $p_0\in X_0$ locally given by the equation $xy-zw=0$. Let $\widetilde{X}\coloneq \operatorname {Bl}_{p_0}(X_0)$ be the blow-up of $X_0$ at the point $p_0$. The exceptional divisor of $p\colon \widetilde{X}\rightarrow X$ is a surface $E$ that is isomorphic to $\mathbb{P}^1\times \mathbb{P}^1$. The surface $E$ in $\widetilde{X}$ can be contracted in two different directions corresponding to the two different projections of $\mathbb{P}^1\times \mathbb{P}^1$. One of the projections $q\colon \widetilde{X}\rightarrow X$ brings us back to the variety $X$ by collapsing the divisor $E$ to the curve $\ell _0$. The second projection $q^+\colon \widetilde{X} \rightarrow X^{+}$ gives a new smooth projective variety $X^+$ and a new curve $\ell _0^+$ for which $K_{X^+}\cdot \ell _0^+=0$. There is a natural projective birational map $\pi \colon X\mathrel{\rightdasharrow }X^+$, i.e., a projective morphism that is defined everywhere at $X$ except at $\ell _0$. Analogously, the inverse $\pi ^{-1}$ is defined everywhere except at $\ell _0^+$. This leads to a commutative diagram

$$\begin{equation*} \vcenter{\img[][92pt][87pt][{$\xymatrix{ & \widetilde{X}\ar[ld]_-{q}\ar[rd]^-{q^+}\ar[dd]^(0.4){p} & \\ X\ar[rd]_-{\phi} \ar@{-->}[rr]^(0.4){\pi} & & X^+\ar[ld]^-{\phi^+} \\ & X_0 & \\ }$}]{Images/imgd03f35a975f4ad4a7a4de43cbf3f2d60.svg}} \end{equation*}$$

The projective contraction $\phi ^+\colon X^+\rightarrow X_0$ is called the flopped contraction while the projective birational map $\pi \colon X \mathrel{\rightdasharrow }X^+$ is called a flop. A flop swaps a $K_X$-trivial curve with a $K_{X^+}$-trivial curve.

In summary, a flop is an algebraic surgery that cuts a curve that intersects the canonical divisor trivially and pastes a new curve that intersects the canonical divisor trivially.

Flips

Let $\phi \colon X \rightarrow Y$ be a flipping contraction. A flip is a birational map $\pi \colon X \mathrel{\rightdasharrow }X^+$ together with a projective morphism $\phi ^+ \colon X^+\rightarrow Y$ such that the following properties hold:

(i)

the divisor $K_{X^+}$ intersects positively every curve contracted by $\phi ^+$, and

(ii)

the equality $\rho (X^+)=\rho (Y)+1$ holds.

Thus, we have a commutative diagram:

$$\begin{equation*} \vcenter{\img[][88pt][47pt][{$\xymatrix{ X\ar@{-->}[rr]^-{\pi}\ar[rd]_-{\phi} & & X^+\ar[ld]^-{\phi^+} \\ & Y & }$}]{Images/img5c24455efed37c5fd0953c0152fbe071.svg}} \end{equation*}$$

The morphism $\phi$ collapses $K_X$-negative curves while the morphism $\phi ^+$ collapses $K_{X^+}$-positive curves. Thus, the composition $\pi =(\phi ^+)^{-1}\circ \phi$ contracts $K_X$-negative curves and extracts $K_{X^+}$-positive curves. This motivates the name of this birational map: it flips the sign of the intersection of the curves with the canonical divisor. It swaps negative curves with positive curves. The morphism $\phi ^+\colon X^+\rightarrow Y$ is called the flipped contraction.

As explained above, the variety $Y$ tends to be too singular to work with it. However, if the flip $\pi$ exists, then $X^+$ exhibits similar singularities to the ones in $X$. In particular, if $K_X$ induces a $\mathbb{Q}$-line bundle, then so does $K_{X^+}$, so the cone theorem and the contraction theorem may still apply to $X^+$. For a long time, proving the existence of the flip $\pi$ of a flipping contraction $\phi$ was one of the most difficult problems on the MMP. This problem was fully settled by the work of Birkar, Cascini, Hacon, McKernan, and Xu, leading to the following theorem (see BCHM10Bir12HX13).

Theorem 7.

Let $X$ be a variety with log canonical singularities and $\phi \colon X\rightarrow Y$ be a flipping contraction. Then, the flip $\pi \colon X\mathrel{\rightdasharrow }X^+$ of $\phi$ exists.

The concept of log canonical singularities is the largest class of singularities in which we expect the MMP to fully work. It will be explained in the next pages. For now, let us emphasize that both the cone theorem and contraction theorem are valid for varieties with log canonical singularities.

Log Pairs

In birational geometry, we often study log pairs instead of algebraic varieties. A log pair is a projective variety $X$ together with a divisor $\Delta$ whose coefficients are all nonnegative,⁠³ such that $K_X+\Delta$ induces a $\mathbb{Q}$-line bundle on $X$. The divisor $\Delta$ is called a boundary divisor.

³

These divisors are called effective.

✖

There are several reasons for which we consider pair structures instead of varieties themselves. For instance, an elliptic curve $E$ admits a group structure induced by its universal cover $\mathbb{C}$, the group $\mathbb{Z}_2$ acts on $E$ by sending $e\mapsto e^{-1}$. The quotient $E/\mathbb{Z}_2$ is isomorphic to the projective line $\mathbb{P}^1$. However, the quotient $\pi \colon E\rightarrow \mathbb{P}^1$ is not free, there are some ramification points. These points correspond to $0$, $\frac{1}{2}$, $\frac{1}{2}i$, and $\frac{1}{2}+\frac{1}{2}i$ in the universal cover $\mathbb{C}$ of $E$. Without loss of generality, we may assume that the images of these points in $\mathbb{P}^1$ are $\{0\},\{1\},\{\infty \}$ and a fourth point $\{\lambda \}$. Consider the boundary divisor $\Delta _{\mathbb{P}^1}=\frac{1}{2}(\{0\}+\{1\}+\{\infty \}+\{\lambda \})$. Then, the divisor

$$\begin{equation} K_{\mathbb{P}^1}+\Delta _{\mathbb{P}^1} {\tag{6}} \end{equation}$$

induces a $\mathbb{Q}$-line bundle on $\mathbb{P}^1$. Both line bundles and $\mathbb{Q}$-line bundles can be pulled-back via morphisms. For instance, a $\mathbb{Q}$-line bundle $\mathcal{L}$ on $\mathbb{P}^1$ induces a $\mathbb{Q}$-line bundle $\pi ^*\mathcal{L}$ on $E$ that associates the line $\mathcal{L}_{\pi (e)}$ to the point $e\in E$. This is called the pull-back of the $\mathbb{Q}$-line bundle. The pull-back to $E$ of the $\mathbb{Q}$-line bundle associated to the divisor 6 equals $\omega _E$. Thus, we write $K_E=\pi ^*( K_{\mathbb{P}^1}+\Delta _{\mathbb{P}^1})$. Hence, studying the canonical divisor $K_E$ of an elliptic curve is analogous to studying the log pair

$$\begin{equation*} \left(\mathbb{P}^1,\frac{1}{2}(\{0\}+\{1\}+\{\lambda \}+\{\infty \}) \right). \end{equation*}$$

This picture generalizes to finite quotients of smooth projective varieties.

The $\mathbb{Q}$-line bundle associated to $K_X+\Delta$ is denoted by $\omega _X(\Delta )$. The concepts of canonically polarized, Calabi–Yau, and Fano varieties extends to log pairs. For instance, a log pair $(X,\Delta )$ is said to be Fano if the sections of $\omega _X^m(m\Delta )$ define an embedding of $X$ into a projective space for some $m<0$.

Singularities of Log Pairs

Let $(X,\Delta )$ be a log pair and $\pi \colon Y\rightarrow X$ be a blow-up. Let $E\subset Y$ be a prime divisor.⁠⁴ The log discrepancy of $(X,\Delta )$ at $E$ is the rational number $1-\operatorname {coeff}_E(\Delta _Y)$, where $\Delta _Y$ is the unique divisor for which

⁴

A prime divisor is a divisor that cannot be written as the sum of two distinct divisors.

✖

$$\begin{equation*} K_Y+\Delta _Y=\pi ^*(K_X+\Delta ). \end{equation*}$$

Roughly speaking, the divisor $E$ in the blow-up represents certain tangent directions on $X$. The log discrepancy $a_E(X,\Delta )$ measures how singular is the pair $(X,\Delta )$ along these tangent directions. The larger this number is, the smoother the pair is in this direction.

We say that a log pair $(X,\Delta )$ is log terminal if all its log discrepancies are positive. We say that a log pair $(X,\Delta )$ is log canonical if all its log discrepancies are nonnegative.

For example, the quotient of a smooth projective variety by a finite group has only quotient singularities which are log terminal. Another construction that leads to log terminal singularities is the cone construction. Let $X\hookrightarrow \mathbb{P}^n$ be an embedding of a projective variety. We may take the homogeneous equations $f_1,\dots ,f_k$ defining $X$ inside $\mathbb{P}^n$ and consider them as equations in $\mathbb{C}^{n+1}$. The affine variety that these equations cut out in $\mathbb{C}^{n+1}$ is called the cone over $X$ with respect to the embedding $X\hookrightarrow \mathbb{P}^n$. This affine variety tends to be singular at the origin. If the embedding $X\hookrightarrow \mathbb{P}^n$ is defined by the sections of $\omega _X^m$ for some $m<0$, we say that the embedding is the m-th anti-pluricanonical embedding. The following theorem allows us to construct several log terminal singularities:

Theorem 8.

Let $X$ be a smooth Fano variety of dimension $n$. The cone of $X$ with respect to an anti-pluricanonical embedding is a log terminal singularity of dimension $n+1$.

For instance, the ordinary double point of dimension $2$, locally given by the equation $\{x^2+y^2+z^2=0\}$, is the cone over the second anti-pluricanonical embedding of $\mathbb{P}^1$. The cones over higher anti-pluricanonical embeddings of $\mathbb{P}^1$ require many more equations to be described. Note that a single smooth Fano variety gives a plethora of log terminal singularities.

In a similar vein, any cone over a smooth Calabi–Yau variety gives a log canonical singularity. An elliptic singularity is a cone over an elliptic curve. It is an example of a log canonical singularity that is not log terminal. For this and some other supporting reasons, it is often said that log terminal singularities are the local version of Fano varieties while log canonical singularities are the local version of Calabi–Yau varieties.

Minimal Model Program

Now we have all the tools to explain what the MMP is. The MMP is both the research topic and the algorithm that we study in this area of research.

The algorithm starts with a log canonical pair $(X,\Delta )$.⁠⁵ If $K_X+\Delta$ intersects every curve nonnegatively,⁠⁶ we declare $(X,\Delta )$ to be a minimal model. This is the outcome of the algorithm. If there is some $(K_X+\Delta )$-negative curve, then by the Cone theorem 5, there must be some extremal $(K_X+\Delta )$-negative curve $C$. By the Contraction theorem 6, there is a contraction $\phi _C$ associated to this extremal negative curve. There are three possibilities for $\phi _C$: divisorial contraction, flipping contraction, or Mori fiber space. If $\phi _C$ induces a Mori fiber space $X\rightarrow Y$, then we again stop the algorithm and declare this to be an outcome. If $\phi _C$ induces a divisorial contraction $X\rightarrow Y$, we replace $X$ with $Y$ and $\Delta$ with its image $\Delta _Y$ on $Y$. The pair $(Y,\Delta _Y)$ is again log canonical, so we can go back to the start of the algorithm. In this case $\rho (Y)=\rho (X)-1$, so this step cannot happen infinitely many times in the algorithm. If $\phi _C\colon X\rightarrow Y$ is a flipping contraction, then neither the cone theorem nor contraction theorem apply to $Y$. However, we may apply Theorem 7 to find a flip $\pi \colon X \mathrel{\rightdasharrow }X^+$. Then, we replace $X$ with $X^+$ and $\Delta$ with its image $\Delta ^+$ in $X^+$. It turns out that $(X^+,\Delta ^+)$ has log canonical singularities. Thus, we may return to the starting point. However, in this case $\rho (X)=\rho (X^+)$. So, the Picard rank alone does not rule out the possibility of iterating this step forever. The flow chart explains the steps of the MMP.

⁵

For simplicity, we can think about a smooth projective variety.

✖

⁶

Divisors with this property are known as numerically eventually free or nef for simplicity.

✖

The flow chart leads to the following.

Theorem 9.

Let $(X,\Delta )$ be a log canonical pair. Then, we can run a $(K_X+\Delta )$-MMP.

In summary, the MMP aims to birationally transform a variety either into a minimal variety or a variety that admits a Mori fiber space. In both cases, the variety acquired some feature that is likely to be helpful to understand its geometry.

Main Conjectures

Theorem 9 states that we can run a minimal model program for any log canonical pair $(X,\Delta )$ in any dimension. Thus, we can perform the algorithm called the Minimal Model Program (MMP). However, this theorem says nothing about this algorithm stopping in finite time. To achieve this, the main conjecture is the following:

Conjecture 1 (Termination of flips).

There is no infinite sequence of flips for a log canonical pair $(X,\Delta ).$

The previous conjecture would imply that the algorithm always terminates. On the other hand, we expect that a minimal model has some extra properties.

Conjecture 2 (Abundance).

Let $(X,\Delta )$ be a log canonical pair with $K_X+\Delta$ nef. Then, there exists some $m>0$ satisfying the following statement. For every $x\in X$ there exists $s\in \Gamma (\omega _X^m(m\Delta ))$ with $s(x)\neq 0$.

The condition described in the previous theorem is known as base point freeness of the line bundle. It implies the existence of a morphism $X\rightarrow Y$ such that every global section of $\omega _X^m(m\Delta )$ is the pull-back of a global section on $Y$. In particular, the variety $Y$ admits the structure of a canonically polarized pair. The variety $Y$ is called an ample model.

In summary, Theorem 9 together with Conjecture 1 and Conjecture 2 imply the following. For any log canonical pair $(X,\Delta )$ there is a birational transformation $(X,\Delta )\mathrel{\rightdasharrow }(X',\Delta ')$ and a morphism $\phi \colon (X',\Delta ') \rightarrow Z$ such that the general fiber of $\phi$ is either: canonically polarized, Calabi–Yau, or Fano. Thus, we would achieve our purpose of reducing the study of varieties to one of the pure classes.

State of the Art

The MMP is fully settled in dimension $3$ due to the work of many mathematicians: Mori, Shokurov, Reid, Kollár, and Kawamata, among others. The termination of flips for log canonical pairs of dimension $3$ is proved in Sho92, while the abundance for log canonical pairs of dimension $3$ is proved in KMM94.

A pair $(X,\Delta )$ is said to be of log general type if the function

$$\begin{equation*} m\mapsto \dim _{\mathbb{C}}\Gamma (\omega _X^m(m\Delta )) \end{equation*}$$

asymptotically behaves as a polynomial of degree $\dim X$. For instance, a canonically polarized log pair is of log general type.

In BCHM10, the authors define the MMP with scaling that is a particular run of the MMP that tends to terminate faster. They prove that the MMP with scaling terminates for log terminal pairs of general type. In particular, we have the following.

Theorem 10.

Let $(X,\Delta )$ be a log terminal pair of log general type. Then, there is some $(K_X+\Delta )$-MMP that terminates with a minimal model.

On the other hand, if the divisor $-(K_X+\Delta )$ has some positivity, then in BCHM10 the authors proved that some MMP terminates with a Mori fiber space.

Theorem 11.

Let $(X,\Delta )$ be a log terminal pair. Assume that $\omega _X^m(m\Delta )$ admits a global section for some $m<0$. Then there is a $(K_X+\Delta )$-MMP that terminates with a Mori fiber space.

Both, Conjecture 1 and Conjecture 2 are open in dimension at least $4$. In the author’s thesis, it is proved that in dimension $4$ termination holds provided that $K_X+\Delta$ satisfies some positivity assumption (see Mor19HM20). In particular, we have the following theorem.

Theorem 12.

Let $(X,\Delta )$ be a log canonical pair of dimension $4$. Assume that $\omega _X^m(m\Delta )$ admits a global section for some $m>0$. Then any $(K_X+\Delta )$-MMP terminates with a minimal model.

Recent and Future Directions

It is common in geometry that the study of smooth geometric objects leads to the study of singular geometric objects. For instance, the study of symmetries of manifolds led to the concept of orbifolds and these became a central topic in topology. In a similar vein, the study of smooth projective varieties naturally leads to the study of log terminal varieties, while the study of moduli spaces of smooth projective varieties leads to the study of log canonical varieties. Indeed, the compactification of moduli spaces of smooth projective varieties tends to parametrize varieties with semi-log canonical singularities. Semi-log canonical singularities are possibly nonnormal singularities whose normalizations have log canonical singularities. There has been a trend in the last few decades to generalize theorems for smooth projective varieties to the setting of either log terminal or log canonical singularities. The author expects that every theorem for smooth projective varieties, after possibly some adequate adjustment, can be generalized to the setting of log canonical pairs.

Results related to the MMP usually appear in three flavors: conjectures surrounding the MMP, applications of the MMP, and the MMP on new categories.

Among the results on conjectures surrounding the MMP in the last few years there have been important results on effectiveness of Iitaka fibrations BZ16, log discrepancies of singularities Mor21, and termination of flips HM20.

One substantial application of the MMP is to the construction of moduli spaces. In order to construct moduli spaces one needs to bound the considered varieties. There are important theorems about boundedness of varieties of general type HMX18, and boundedness of Fano varieties Bir21b. The author expects that the tools of the MMP will lead to important boundedness theorems on Calabi–Yau varieties in the next few years. The MMP has played a role in the construction of the moduli spaces of canonically polarized varieties. The compactifications of these moduli spaces, that parametrize semi-log canonical varieties, are known as KSBA moduli. On the other hand, the MMP has played a role in the construction of the moduli spaces of K-semistable Fano varieties known as K-moduli. Recently, Kollár published a book that gives a complete treatment of the moduli of canonically polarized varieties Kol23. On the other hand, the survey Xu21 explains several important results on K-stability and the existence of K-moduli spaces.

The MMP has been heavily used to study Fano varieties: global sections of antipluricanonical systems Bir19, boundedness of Fano vareities Bir21b, and existence of Kähler–Einstein metrics Xu21. In these three directions, the study of log Calabi–Yau structures on Fano varieties has been a central topic.

Finally, the MMP has been generalized to different categories: Kähler complex manifolds HP16, positive and mixed characteristics HW23, generalized pairs Bir21a, and foliated varieties CS21. In the upcoming years, many theorems and conjectures of the MMP will be solved in these new categories. When proving the theorems of the MMP in these new categories many new techniques are discovered and tricks introduced. Some of them reprove known theorems in the MMP. The further development of the MMP in these new settings is expected to also shed light on the classic conjectures of the program: termination of flips and abundance.