Fact-checked by Grok 2 weeks ago

Birational geometry

Birational geometry is a branch of algebraic geometry that studies algebraic varieties up to birational equivalence, where two varieties are considered equivalent if there exists a rational map between them that induces an isomorphism between their function fields or, equivalently, is an isomorphism on dense open subsets.^[1] This equivalence ignores "small" subsets of codimension at least two, allowing focus on intrinsic properties preserved under such transformations, such as the geometry of the function field.^[2] The primary goal of birational geometry is to classify algebraic varieties by constructing simplified models through birational transformations, primarily via the Minimal Model Program (MMP), which systematically reduces complexity using operations like contractions of extremal rays and flips.^[3] Key invariants include the Kodaira dimension, which measures the growth rate of pluricanonical sections and categorizes varieties from -∞ (uniruled) to the dimension of the variety (general type), guiding the MMP toward minimal models with nef canonical divisors.^[1] The MMP also involves resolving singularities to terminal or canonical types, ensuring Q-factoriality for well-behaved birational maps.^[3] Historically, birational geometry traces back to classical efforts in the 19th and early 20th centuries to understand rational maps and rationality of varieties, but it was revolutionized in the 1980s by Shigefumi Mori's theory of extremal rays on the Mori cone, enabling contractions and the modern MMP framework.^[3] Landmark progress came with the 1998 book Birational Geometry of Algebraic Varieties by János Kollár and Shigefumi Mori, providing foundational tools for higher-dimensional cases, particularly flips in dimension three.^[3] The program was fully established in characteristic zero in 2010 by the theorem of Caucher Birkar, Paolo Cascini, Christopher Hacon, and James McKernan, proving the existence of minimal models for varieties of log general type and resolving long-standing conjectures on flips and abundance; Birkar's contributions, including this theorem, earned him the Fields Medal in 2018.^[4]^[5] Beyond classification, birational geometry has profound applications in arithmetic geometry, such as studying rational points on varieties via conjectures like those of Lang and Vojta, and in moduli theory, where it elucidates the birational structure of spaces parametrizing curves or sheaves.^[1] Recent extensions include work on positive characteristic and stacks, broadening its scope to Deligne-Mumford stacks and equivariant settings.^[2]

Birational Maps and Equivalence

Rational Maps

In algebraic geometry, a rational map between two varieties X and Y defined over an algebraically closed field k is an equivalence class of pairs (U, f), where U \subset X is a dense open subset and f: U \to Y is a morphism of varieties, with two such pairs equivalent if they agree on the intersection of their domains, which is again dense open.^[6]^[7] This definition captures maps that are "defined almost everywhere" on X, allowing for points of indeterminacy where the map cannot be extended regularly.^[8] A classic example is the projection from the affine plane \mathbb{A}^2_k to \mathbb{A}^1_k given by (x, y) \mapsto x/y, which is defined on the dense open set where y \neq 0, but indeterminate at the origin (0,0).^[6] In the projective setting, a rational map from \mathbb{P}^n_k to \mathbb{P}^m_k can be given by homogeneous polynomials of the same degree, such as the pair of linear forms defining [x:y:z] \mapsto [x:y] on \mathbb{P}^2_k, which is a morphism except at the point [0:0:1] where both forms vanish simultaneously, resulting in indeterminacy.^[8] The indeterminacy locus of a rational map is the complement of its maximal domain of definition, a proper closed subset of X consisting of points where no representative morphism is regular.^[7] Rational maps compose whenever the image of the first lies in the domain of the second, specifically on the dense open set where their domains overlap, yielding another rational map.^[6] A rational map is a morphism if and only if it is regular everywhere on X, meaning its domain of definition is all of X.^[8] For projective varieties, any rational map X \dashrightarrow Y with Y embedded in \mathbb{P}^N_k is uniquely determined by a linear system of divisors on X, namely the complete linear system |D| associated to a line bundle whose global sections define the map via the projective embedding.^[9] This correspondence underscores the role of linear systems in classifying rational maps between projective varieties.^[9]

Birational Maps

In algebraic geometry, a birational map between two irreducible varieties X and Y over a field k is a rational map \phi: X \dashedrightarrow Y that admits a rational inverse \psi: Y \dashedrightarrow X such that the compositions \psi \circ \phi: X \dashedrightarrow X and \phi \circ \psi: Y \dashedrightarrow Y are equal to the identity map on some dense open subsets of X and Y, respectively.^[10] This definition specializes the broader notion of rational maps, requiring invertibility in the sense of rational correspondences.^[10] Birational maps exhibit key properties that highlight their role in studying varieties up to "rational equivalence." Specifically, any birational map \phi: X \dashedrightarrow Y induces an isomorphism between dense open subsets U \subset X and V \subset Y, where \phi restricts to a regular isomorphism U \to V.^[11] Furthermore, birational maps preserve the function fields of the varieties, inducing a k-isomorphism k(X) \cong k(Y) between the fields of rational functions on X and Y.^[12] This isomorphism arises because the generic points of X and Y map to each other under the rational correspondence, equating the residue fields at those points.^[12] A central theorem in the subject characterizes birationality intrinsically via function fields: two irreducible varieties X and Y over a field k are birational if and only if their function fields k(X) and k(Y) are isomorphic as extensions of k.^[13] This equivalence underscores the function field as the primary birational invariant, allowing classification problems to be reformulated in terms of field theory. As rational maps, birational maps may suffer from indeterminacy at certain points or loci where they cannot be evaluated continuously. To resolve such indeterminacies and obtain a regular extension, one can perform blow-ups along suitable centers (such as points or subvarieties) in the domain variety; the resulting morphism from the blow-up is then a proper birational map that agrees with the original on the complement of the exceptional locus.^[14] Successive blow-ups may be required to fully resolve the indeterminacy locus, particularly in higher dimensions.^[14] The concept of birational maps was introduced by Max Noether in the context of Cremona transformations of the projective plane, where he studied their generation and factorization properties.^[15]

Birational Equivalence

Two algebraic varieties X and Y over an algebraically closed field k are said to be birational, denoted X \sim Y, if there exists a birational map f: X \dashrightarrow Y. This relation is an equivalence relation on the set of varieties, as it is reflexive (via the identity map), symmetric (by inverting the birational map), and transitive (by composing birational maps along a chain). Birational varieties share the same function field k(X) \cong k(Y), which is the field of rational functions on X (or Y), consisting of quotients of regular functions where the denominator is non-zero on a dense open set. Equivalently, X and Y are birational if they contain isomorphic Zariski-open subsets, meaning they agree on dense open subsets and differ only on lower-dimensional loci.^[16]^[17]^[18] A key application of birational equivalence is the notion of rationality: a variety X is rational if it is birational to projective space \mathbb{P}^n_k for n = \dim X. Rational varieties admit a parametrization by rational functions, facilitating the study of their geometry via coordinates on \mathbb{P}^n. A classical question in this context is the rationality of hypersurfaces; for instance, smooth plane conics are rational (via projection from a point on the curve), but smooth plane cubics are elliptic curves and not rational, while the rationality of smooth hypersurfaces of degree d \geq 3 in \mathbb{P}^n for n \geq 3 remains open in general, with counterexamples known only in specific cases like certain quartic surfaces. This problem, often termed Noether's problem in the birational classification of hypersurfaces, highlights the challenges in determining birational type beyond low dimensions.^[16]^[19] Birational equivalence preserves several fundamental properties of varieties. The dimension \dim X = \operatorname{tr.deg}_k k(X) is invariant, as it equals the transcendence degree of the shared function field. Irreducibility is also preserved, since the function field of an irreducible variety is a field, whereas reducible varieties have function fields that are products of fields corresponding to irreducible components. For smooth proper varieties, birational maps induce isomorphisms on the N'eron-Severi group \mathrm{NS}(X) = \mathrm{Pic}(X)/\mathrm{Tors}, the Picard group modulo torsion, reflecting the birational invariance of line bundles up to algebraic equivalence.^[16]^[17] In characteristic zero, every variety admits a smooth proper birational model: Hironaka's theorem guarantees the existence of a proper birational morphism \pi: \tilde{X} \to X from a smooth variety \tilde{X} to X, obtained via a finite sequence of blow-ups along smooth centers, resolving singularities while preserving the function field. Such models are not unique, but they provide canonical representatives within a birational equivalence class, enabling the study of birational properties in a smooth projective setting. This resolution underpins much of modern birational geometry, allowing normalization and minimal model constructions.^[16]^[17]

Examples of Birational Transformations

Plane Conics

A plane conic is defined as the zero locus in the projective plane \mathbb{P}^2 of a homogeneous quadratic polynomial Q(x,y,z) = a x^2 + b x y + c y^2 + d x z + e y z + f z^2 = 0.^[20] The conic is smooth if the associated symmetric matrix has full rank 3, equivalently if its discriminant \Delta = \begin{vmatrix} a & b/2 & d/2 \\ b/2 & c & e/2 \\ d/2 & e/2 & f \end{vmatrix} \neq 0.^[21] Over an algebraically closed field, smooth plane conics are all isomorphic to the projective line \mathbb{P}^1, and thus serve as a fundamental example of rationality in birational geometry.^[22] An explicit birational equivalence between a smooth conic C and \mathbb{P}^1 is given by stereographic projection from a chosen point p \in C. This map sends a point q \in C \setminus \{p\} to the second intersection point of the line \overline{pq} with the line at infinity, yielding a rational map \pi: C \dashrightarrow \mathbb{P}^1 that is an isomorphism away from p.^[23] For instance, on the conic x^2 + y^2 - z^2 = 0 with p = (0:1:1), the projection can be expressed in affine coordinates as (x/z, y/z) \mapsto t = x/(1 - y), with inverse given by rational functions in t. This construction highlights how birational maps preserve the function field k(C) \cong k(t), confirming the rationality of smooth conics.^[22] In the singular case, where \Delta = 0, the conic degenerates: if the matrix has rank 2, it consists of two distinct lines intersecting at a node; if rank 1, it is a double line, which can be viewed as cuspidal in its scheme-theoretic structure.^[21] The normalization of a nodal conic yields two disjoint copies of \mathbb{P}^1, one for each line, while the normalization of the cuspidal (double line) case is a single \mathbb{P}^1 with a degree-2 map to the conic.^[24] In both instances, the components or normalized model are birational to \mathbb{P}^1, underscoring that all plane conics are rational varieties.^[25] A rational parametrization of a conic can be constructed using a parameter t \in \mathbb{P}^1 via lines through a base point. For the general equation a x^2 + b x y + c y^2 + d x z + e y z + f z^2 = 0, assuming a point such as (1:0:0) lies on it (or adjusting accordingly), the map is given by

(x:y:z) = \left(1 : t : \frac{- (a + b t + c t^2)}{d + e t + f t^2}\right),

which traces the conic rationally except possibly at points where the denominator vanishes.^[22] This parametrization extends to singular cases by resolving the indeterminacies along the components. Historically, the study of plane conics and their transformations laid foundational groundwork for birational geometry in the 19th century, with Arthur Cayley advancing the classification of conics through invariants of binary quadratic forms and exploring rational correspondences between them.^[26] Cayley's work on the moduli and equivalence of conics under projective transformations influenced early efforts to understand birational invariants, bridging classical projective geometry with abstract algebraic approaches.^[27]

Quadrics and Projective Spaces

In algebraic geometry, a smooth quadric hypersurface Q^n \subset \mathbb{P}^{n+1} over an algebraically closed field of characteristic not equal to 2 is defined by the equation \sum_{i=0}^{n+1} x_i^2 = 0.^[28] This variety is rational, meaning it is birational to the projective space \mathbb{P}^n.^[28] The birational equivalence holds because the quadric contains rational points over such fields, allowing a stereographic projection that establishes the map.^[28] The birational map from Q^n to \mathbb{P}^n is constructed via projection from a point p on the quadric to a hyperplane not containing p. For instance, taking p = (1:i:0:\dots:0) (where i = \sqrt{-1}), which lies on Q^n over \mathbb{C}, the projection \pi_p: Q^n \dashrightarrow H to the hyperplane H = \{x_0 = 0\} \cong \mathbb{P}^n is birational, as the inverse is obtained by lines through p intersecting H.^[29] This projection is undefined only at p, but the map extends rationally and is an isomorphism away from the lines through p tangent to Q^n. For n=3, the quadric threefold Q^3 \subset \mathbb{P}^4, this parametrizes points via lines through p, yielding a birational equivalence to \mathbb{P}^3. For the specific case of a quadric surface (n=2, Q^2 \subset \mathbb{P}^3), the variety is isomorphic to \mathbb{P}^1 \times \mathbb{P}^1, which provides an explicit parametrization using its two families of rulings—lines covering the surface.^[30] The isomorphism is given by the Segre embedding \phi: \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^3, [x_0:x_1], [y_0:y_1] \mapsto [x_0 y_0 : x_0 y_1 : x_1 y_0 : x_1 y_1], for the equation z_0 z_3 - z_1 z_2 = 0.^[30] The inverse map sends [z_0:z_1:z_2:z_3] \mapsto ([z_0:z_1], [z_2:z_3]) where defined, confirming the birational (in fact, isomorphic) relation to \mathbb{P}^2 via projection from a point on one ruling. The rulings facilitate this by parametrizing the surface as a product, birationally equivalent to projective plane. Singular quadrics over algebraically closed fields are also rational and birational to \mathbb{P}^n. For example, a quadric cone (with 0-dimensional vertex) is birational to \mathbb{P}^n via parametrizations using its rulings or by blowing up the singular locus to obtain a \mathbb{P}^1-bundle over a smooth quadric of dimension n-1, which is rational, making the total space rational as well.^[28] This extends the smooth case while preserving rationality. As the dimension-1 analog, plane conics are birational to \mathbb{P}^1.^[30]

Resolution of Singularities

Techniques for Resolution

The primary technique for resolving singularities in algebraic geometry is the blow-up operation along a subvariety, which constructs a birational morphism from a new variety to the original one, replacing singular points with projective bundles over the exceptional locus.^[31] Given a variety X and a closed subvariety Z \subset X, the blow-up \mathrm{Bl}_Z X is defined as the Proj of the Rees algebra associated to the ideal sheaf of Z, resulting in a morphism \pi: \mathrm{Bl}_Z X \to X that is an isomorphism away from Z and whose fiber over points in Z is the projectivized normal cone to Z in X.^[32] The exceptional divisor E = \pi^{-1}(Z) is a Cartier divisor isomorphic to the projectivized normal bundle \mathbb{P}(\mathcal{N}_Z X), which smooths singularities by separating directions in the tangent space at those points.^[31] A landmark result establishing the effectiveness of blow-ups for resolution is Hironaka's theorem, which asserts that every algebraic variety over a field of characteristic zero admits a resolution of singularities via a finite sequence of blow-ups along smooth centers. This theorem, proved in two parts, shows that starting from any singular variety X, one can iteratively select smooth subvarieties (centers) contained in the singular locus and blow up along them, eventually obtaining a smooth variety \tilde{X} with a proper birational morphism \pi: \tilde{X} \to X that is an isomorphism over the smooth points of X. The process terminates because each blow-up reduces the multiplicity of the singular locus in a controlled manner, ensuring no infinite descent occurs in characteristic zero.^[33] Beyond general blow-ups, specialized techniques exist for certain classes of varieties. For curves, normalization provides a one-step resolution: given a singular curve C, its normalization \tilde{C} is the unique smooth curve birational to C that separates branches at singular points, achieved by mapping the integral closure of the coordinate ring of C.^[34] In the toric setting, singularities of toric varieties—defined by fans in a lattice—can be resolved by subdividing the fan into a smooth fan via stellar subdivisions, yielding a toric resolution that is a composition of toric blow-ups along torus-invariant subvarieties. For pairs (X, D) consisting of a variety X and a divisor D, a log resolution is a birational morphism \pi: Y \to X from a smooth Y such that both the strict transform \pi^{-* }D and the exceptional locus are normal crossing divisors, often obtained by blowing up along strata of the pair to handle logarithmic singularities. Resolution techniques are distinguished by whether they are embedded or abstract, and involve the notion of strict transforms to track images of subvarieties. An embedded resolution resolves singularities within an ambient smooth variety, such as blowing up to make a singular subvariety have normal crossings with the ambient space, whereas an abstract resolution directly smooths the variety without reference to an embedding.^[35] The strict transform of a subvariety V \subset X under a blow-up \pi: \tilde{X} \to X is the closure of \pi^{-1}(V \setminus Z) in \tilde{X}, excluding components mapped to the center Z, which allows iterative resolution by focusing on the transformed singular locus.^[32] A concrete example illustrates blow-up resolution: consider a nodal curve C \subset \mathbb{A}^2 defined by xy = 0, which has a singularity at the origin where two branches cross transversely. Blowing up \mathbb{A}^2 at the origin yields \mathrm{Bl}_0 \mathbb{A}^2, isomorphic to the total space of \mathcal{O}_{\mathbb{P}^1}(-1), and the strict transform \tilde{C} of C consists of two disjoint smooth lines meeting the exceptional divisor \mathbb{P}^1 at distinct points, thus resolving the node.^[36]

Impact on Birational Properties

Resolution of singularities preserves fundamental birational invariants of algebraic varieties. For a resolution morphism \pi: Y \to X, where Y is a smooth variety birational to the possibly singular variety X, the function fields satisfy K(X) \cong K(Y), as birational equivalence is defined by the isomorphism of their fields of rational functions.^[17] Likewise, the Kodaira dimension \kappa(X) = \kappa(Y), since it depends only on the growth of dimensions of spaces of sections of powers of the canonical sheaf and is thus invariant under birational maps between smooth projective varieties.^[37] Although birational invariants remain unchanged, the resolution alters certain topological properties. The exceptional divisors, which are the components of \pi^{-1}(S) where S is the singular locus of X, introduce additional structure that modifies invariants like the Euler characteristic. Specifically, \chi(Y, \mathcal{O}_Y) generally differs from \chi(X, \mathcal{O}_X) because the topology of the exceptional locus contributes non-trivially to the computation, as seen in formulas relating the zeta-function or Alexander polynomial of singularities to the Euler characteristics of the smooth parts of these divisors.^[38] A key result is that the resolved variety Y is birational to the original X, ensuring they share the same birational type. In characteristic zero, Hironaka's theorem guarantees the existence of such a resolution for any variety, and any two smooth proper models of X are birational to each other, allowing a unified study within the birational equivalence class.^[37] This uniqueness up to birational equivalence facilitates classification efforts, as resolutions enable the application of tools from smooth geometry—such as Hodge theory or vanishing theorems—to infer properties of the original singular variety.^[37] Resolutions thus play a pivotal role in classifying varieties by birational type, shifting focus to smooth representatives where invariants like the fundamental group or Hodge numbers can be more readily computed and compared. However, in positive characteristic, these processes face significant limitations: the existence of resolutions is not assured, remaining an open problem for embedded resolutions of varieties of dimension greater than three. While Artin's approximation theorem allows formal solutions to equations defining singularities to be approximated by algebraic ones, it does not generally yield a global smooth birational model.^[39]^[40]

Minimal Model Program

Surfaces and Low Dimensions

In the classical minimal model program for algebraic surfaces over the complex numbers, each birational equivalence class admits a unique minimal model, which is a smooth projective surface with nef canonical divisor—meaning the canonical class intersects non-negatively with every irreducible curve on the surface. This minimal model serves as a canonical representative for the class, facilitating classification by eliminating superfluous exceptional divisors. The nef condition ensures that no further contractions of (-1)-curves are possible, as such curves would violate the non-negativity.^[41] Castelnuovo's theorem, established in 1900, proves that every smooth projective surface is birationally equivalent to a unique minimal surface, and the process of obtaining it involves iteratively contracting exceptional curves of the first kind—irreducible rational curves with self-intersection -1—via blow-down maps until none remain. These contractions preserve the function field and reduce the topological complexity, such as the Euler characteristic, while maintaining birational equivalence. The theorem underpins the entire classification effort for surfaces by guaranteeing the existence and uniqueness of this simplified model.^[41] The process to construct the minimal model typically begins with a resolution of singularities if the original surface is singular, yielding a smooth birational model, followed by successive contractions of all (-1)-curves. Each such contraction replaces the curve with a point, resulting in a surface where the canonical divisor is nef, and the model is unique up to isomorphism within the birational class.^[41] This framework leads to the Enriques-Kodaira classification of minimal surfaces, which organizes them into categories based on the Kodaira dimension—a birational invariant measuring the growth of plurigenera—and additional topological invariants like the second Betti number. The classes include: rational surfaces (Kodaira dimension -\infty, birationally equivalent to \mathbb{P}^2); K3 surfaces (Kodaira dimension 0, with trivial canonical bundle and b_2 = 22); abelian surfaces (Kodaira dimension 0, complex tori of dimension 2); Enriques surfaces (Kodaira dimension 0, quotients of K3 surfaces by fixed-point-free involutions, with b_2 = 10 and no global 2-torsion in H^1); bielliptic surfaces (Kodaira dimension 0, quotients of abelian surfaces by finite group actions); elliptic surfaces (Kodaira dimension 1, fibrations over curves with elliptic fibers); and surfaces of general type (Kodaira dimension 2, with ample canonical bundle on the minimal model). This classification exhaustively covers all minimal models and highlights their geometric diversity.^[42] Iitaka's program builds on this foundation by introducing a fibration method to construct canonical models from minimal ones, particularly for surfaces with positive Kodaira dimension: it resolves the base locus of the canonical linear system to obtain a fibration over a curve, whose general fiber reflects the structure of the canonical ring and previews the higher-dimensional minimal model program.^[43]

Higher Dimensions and Flips

The minimal model program (MMP) in dimensions three and higher aims to classify algebraic varieties up to birational equivalence by constructing minimal or canonical models through a sequence of birational operations. These operations include blow-ups to resolve singularities, divisorial contractions that map exceptional divisors to lower-dimensional loci such as points or curves, flips as small birational modifications, and fiber space contractions that reduce the relative dimension by contracting families of curves.^[44] The process targets a model where the canonical divisor K_X + \Delta is either nef (minimal model) or ample (canonical model) for a pair (X, \Delta) with \Delta an effective \mathbb{Q}-divisor, building on the foundational results for surfaces but requiring new tools like flips to handle the increased complexity of higher-dimensional geometry. Flips are birational maps f: X \dashrightarrow X^+ that contract an exceptional divisor E \subset X to a point while simultaneously expanding another divisor F \subset X^+ from a point, preserving the canonical class in the sense that K_{X^+} + f_* \Delta = f_*(K_X + \Delta). More precisely, a (K_X + \Delta)-flip arises from a small birational morphism (flipping contraction) where -(K_X + \Delta) is f-ample on the exceptional locus, ensuring the flip maintains log canonical singularities and \mathbb{Q}-factoriality.^[44] These operations are essential in higher dimensions because simple contractions may lead to singularities that cannot be resolved without altering the canonical class, unlike in surface theory where contractions suffice. Mori's program, developed in the 1980s, provides the theoretical backbone by focusing on contractions of extremal rays in the cone of effective curves \overline{\mathrm{NE}}_1(X). The contraction theorem asserts that for any extremal ray R on a projective variety X with \mathbb{Q}-factorial terminal singularities, there exists a contraction morphism \phi: X \to Y such that \phi_* R = 0 and every curve contracted by \phi is in R. This allows the MMP to systematically reduce the cone until K_X + \Delta becomes nef or leads to a Mori fiber space structure. The program was first completed in dimension three by Shokurov, who proved the existence of 3-fold log flips for klt pairs, ensuring the MMP terminates with finitely many steps.^[45] In higher dimensions, the log minimal model program (LMMP) extends this via the work of Birkar, Cascini, Hacon, and McKernan, who established the existence of flips and minimal models for varieties of log general type in arbitrary dimension over fields of characteristic zero.^[4] Their results imply that for any klt pair (X, \Delta) with K_X + \Delta pseudo-effective, a log terminal model exists, and the MMP with scaling terminates after finitely many flips. Recent progress in the 2020s includes proofs of termination for pseudo-effective 4-fold flips, advancing boundedness results for singularities in the MMP.^[46] Thus, in characteristic zero, every smooth projective variety admits either a minimal model or a Mori fiber space via the MMP, with minimal models existing for those of log general type, confirming key conjectures of the higher-dimensional MMP.^[4]

Birational Invariants

Kodaira Dimension

The Kodaira dimension of a smooth projective variety X over \mathbb{C}, denoted \kappa(X), is defined as the Iitaka dimension of its canonical divisor K_X, given by

\kappa(X) = \limsup_{m \to \infty} \frac{1}{m} \log \dim H^0(X, mK_X),

where the plurigenera P_m(X) = \dim H^0(X, mK_X) measure the growth of sections of powers of the canonical bundle; if P_m(X) = 0 for all m \geq 1, then \kappa(X) = -\infty.^[37] This definition captures the asymptotic behavior of the canonical ring and serves as a fundamental birational invariant in the classification of varieties.^[47] The possible values of \kappa(X) range from -\infty to \dim X. Varieties with \kappa(X) = -\infty are uniruled, admitting a dominant rational map from \mathbb{P}^N \times C for some curve C, as the canonical bundle restricts negatively on rational curves covering X.^[47] Those with \kappa(X) = 0 include Calabi--Yau varieties such as K3 surfaces and abelian varieties, which are not of general type but exhibit bounded plurigenera growth.^[37] Varieties of general type satisfy \kappa(X) = \dim X, indicating positive canonical growth, while intermediate values arise in fibrations, such as elliptic fibrations over a base Y where \kappa(X) = \kappa(Y). Rational varieties like projective space have \kappa = -\infty, contrasting with varieties of general type.^[37] The Kodaira dimension is birationally invariant: for smooth projective varieties X and Y that are birationally equivalent, \kappa(X) = \kappa(Y), along with equality of all plurigenera P_m(X) = P_m(Y).^[47] It is also monotonic under surjective morphisms with connected fibers f: X \to Y, satisfying \kappa(X) \geq \kappa(Y); more precisely, for a fibration with general fiber F, the inequality \kappa(X) \geq \kappa(F) + \kappa(Y) holds.^[37] The Iitaka fibration theorem refines this: if \kappa(X) = r > 0, there exists a rational map, the Iitaka map \phi_{|mK_X|}: X \dashrightarrow Z for large m, whose image Z has dimension r = \kappa(X), and the general fiber has \kappa = 0, with the map birational onto its image.^[37] In particular, \kappa(X) equals the dimension of the image of the Iitaka map.^[47] Computations of \kappa(X) are explicit in low dimensions and tie into minimal model theory. For smooth projective curves C of genus g, \kappa(C) = -\infty if g=0 (rational curves), \kappa(C) = 0 if g=1 (elliptic curves), and \kappa(C) = 1 if g \geq 2, reflecting the degree of the canonical bundle $2g-2.^[47] For surfaces, resolution of singularities followed by contraction of (-1)-curves yields a minimal model, on which \kappa(S) is determined by the canonical system: \kappa(S) = -\infty for ruled surfaces, \kappa(S) = 0 for Enriques or K3 surfaces (where P_1(S) = 1 and higher plurigenera vanish), \kappa(S) = 1 for minimal elliptic surfaces over a base of positive genus, and \kappa(S) = 2 for surfaces of general type, aligning with the classification via invariants like the Noether inequality.^[37]

Plurigenera

In birational geometry, the k-th plurigenus of a smooth projective variety X over a field of characteristic zero is defined as p_k(X) = h^0(X, kK_X), where K_X denotes the canonical divisor of X.^[7] This measures the dimension of the space of global sections of the k-th power of the canonical sheaf. The plurigenera generalize the geometric genus p_g(X) = p_1(X) to higher multiples and play a central role in classifying varieties up to birational equivalence. Moreover, the plurigenera are birational invariants: if X and Y are smooth projective varieties birational over the base field, then p_k(X) = p_k(Y) for all k \geq 0. This follows from the isomorphism of the canonical sheaves on dense open sets and extension properties under resolution of singularities.^[7] Representative examples illustrate these invariants. For projective space \mathbb{P}^n, the canonical divisor is - (n+1)H where H is the hyperplane class, so p_k(\mathbb{P}^n) = 0 for all k \geq 1.^[7] For a smooth projective curve C of genus g \geq 2, the Riemann-Roch theorem yields p_k(C) = k(2g-2) - g + 1 for k \geq 1, reflecting linear growth tied to the degree of the canonical bundle.^[7] For Enriques surfaces, p_k(X) = 1 if k is even and $0 if k is odd, showing that the sequence can decrease. Asymptotically, the plurigenera exhibit growth \log p_k \sim \kappa \log k, where \kappa is a non-negative integer invariant; this growth rate briefly informs the Kodaira dimension, though the full sequence \{p_k\} provides finer birational data.^[7] In the minimal model program, relative plurigenera h^0(X/B, kK_{X/B}) over a base B guide contractions and flips by controlling the relative canonical growth. A key theorem states that for minimal models of varieties of log general type, the canonical ring is finitely generated, implying that the plurigenera stabilize in the sense of eventual polynomial growth: there exists N such that for k \geq N, p_k(X) equals a polynomial in k of degree equal to the dimension of X.

Hodge Numbers and Fundamental Groups

In birational geometry over the complex numbers, the Hodge numbers h^{p,0}(X) = \dim_{\mathbb{C}} H^0(X, \Omega^p_X) serve as key birational invariants for smooth projective varieties, where \Omega^p_X denotes the sheaf of holomorphic p-forms on X. These numbers capture the dimensions of global sections of the sheaf of differentials and are invariant under birational equivalence due to extension properties like Hartogs' theorem, which allow sections on dense open sets to extend across codimension-one subsets.^[48] A resolution of singularities, if needed, preserves these invariants, ensuring that h^{p,0}(X) = h^{p,0}(Y) for birationally equivalent smooth projective varieties X and Y. By Serre duality, the numbers h^{0,q}(X) are also birational invariants. The étale fundamental group \pi_1^{\ét}(X) provides another topological birational invariant for smooth projective varieties over \mathbb{C}. This profinite group, which classifies finite étale covers of X, is independent of the choice of smooth model within a birational class, as established in the foundational work on étale cohomology. Over \mathbb{C}, the comparison theorems of Deligne link the étale fundamental group to the topological fundamental group, confirming its birational invariance via the isomorphism between étale and singular cohomology.^[49] Illustrative examples highlight these invariants' utility. For rational varieties, birational to projective space \mathbb{P}^n, the étale fundamental group is trivial, reflecting the simply connected nature of \mathbb{P}^n.^[49]

Special Classes of Varieties

Uniruled Varieties

In algebraic geometry, a projective variety X over an algebraically closed field of characteristic zero is defined to be uniruled if there exists a projective variety Y and a dominant rational map \mathbb{P}^1 \times Y \dashrightarrow X.^[50] This condition implies that X admits a dense covering family of rational curves, meaning that the deformations of these curves sweep out a Zariski-open dense subset of X. Equivalently, there exists a rational curve through a general point of X whose deformations cover a dense open subset.^[50] Uniruled varieties exhibit specific birational properties, notably that their Kodaira dimension satisfies \kappa(X) = -\infty. A stronger notion is that of rationally connected varieties, where for any two general points x, y \in X, there exists a rational curve connecting them; every rationally connected variety is uniruled, but the converse does not hold in general. The negative Kodaira dimension serves as an indicator linking uniruledness to the growth rate of pluricanonical systems. A fundamental characterization is given by the following theorem: for a smooth projective variety X over \mathbb{C}, X is uniruled if and only if the canonical divisor K_X is not nef. This equivalence, proved independently by Miyaoka and by Mori in 1987, relies on the existence of a rational curve C through a general point x \in X such that K_X \cdot C < 0. The "if" direction follows from the fact that non-nef K_X implies the existence of such negative curves, while the "only if" direction uses deformation theory to show that uniruledness forces K_X to intersect some curve negatively. Classic examples of uniruled varieties include projective space \mathbb{P}^n, which is covered by lines; smooth quadric hypersurfaces in \mathbb{P}^{n+1}, parametrized by rational curves of low degree; flag varieties such as the variety of complete flags in \mathbb{C}^{n+1}; and Grassmannians \mathrm{Gr}(k, n), which admit families of rational curves through general points via Schubert cycles. These homogeneous spaces are in fact rationally connected, illustrating the geometric abundance of rational curves in such settings. The bend-and-break technique, introduced by Mori in 1979, is a key method for establishing the existence of rational curves on uniruled varieties. Given a curve of non-negative genus passing through specified points on X with K_X not nef, the technique deforms the curve in a suitable Hilbert scheme while fixing the points, leading to a "breaking" into a connected chain of rational curves upon specialization. This deformation-theoretic argument produces free rational curves and underpins proofs of the uniruledness criterion by iteratively applying the lemma to extremal rays in the Mori cone.

Fano Varieties

A Fano variety is defined as a smooth projective variety X over the complex numbers such that the anticanonical divisor -K_X is ample.^[51] The index r of X is the largest positive integer such that -K_X = rH for some ample Cartier divisor H on X.^[52] This notion generalizes projective spaces and quadrics, where the index equals the dimension plus one. Fano varieties form an important subclass of uniruled varieties, characterized by the ampleness of -K_X. Fano varieties exhibit strong connectivity properties: they are uniruled, meaning they admit a dominant rational map from a product of the projective line with X, and rationally connected, meaning any two general points can be joined by a chain of rational curves.^[53] A fundamental result due to Mori establishes that every Fano variety contains rational curves through any general point, ensuring a rich supply of such curves that deform freely and cover the variety.^[54] Many Fano varieties, particularly those arising in classifications, have Picard number one, meaning the Néron-Severi group is generated by a single ample divisor.^[55] In dimension two, Fano varieties are precisely the del Pezzo surfaces, which are classified into ten deformation types: \mathbb{P}^2, \mathbb{P}^1 \times \mathbb{P}^1, and blow-ups of \mathbb{P}^2 at up to eight points in general position.^[56] For threefolds, the classification is partial: Iskovskikh provided a complete list for smooth Fano threefolds of Picard rank one and index at least two, while Prokhorov extended results to certain singular cases and families with higher Picard rank, yielding 105 deformation families in total for smooth examples.^[57]^[58] In higher dimensions, a full classification remains open, but the minimal model program implies boundedness: there are only finitely many deformation types of Fano varieties of a given dimension with at most klt singularities and anticanonical degree bounded below, as established by Birkar, Cascini, Hacon, and McKernan.^[59] Recent advances focus on stability conditions for moduli problems. Odaka and others developed criteria for K-stability of Fano varieties using the alpha-invariant, showing that if \alpha(X) > \dim X / (\dim X + 1), then (X, -K_X) is K-stable, enabling the construction of moduli spaces via geometric invariant theory.^[60] As of 2025, significant progress has been made in constructing explicit moduli spaces of K-polystable Fano varieties, including classifications of one-dimensional components in certain deformation families and studies of birational rigidity using alpha invariants.^[61]^[62] These developments link birational geometry with stability, providing tools to study families of Fanos beyond classical classifications.^[63]

Birational Automorphism Groups

Definition and Examples

The birational automorphism group of an algebraic variety X over a field k, denoted \Bir(X), is the group consisting of all birational self-maps of X, equipped with composition as the group operation.^[64] These maps are rational maps that admit inverses which are also rational maps, and \Bir(X) serves as a fundamental birational invariant of X.^[64] A primary example is the Cremona group \Cr_n(k) = \Bir(\mathbb{P}^n_k), which comprises all birational automorphisms of n-dimensional projective space over k.^[15] In dimension n=2, Noether's theorem (extended by Castelnuovo) states that \Cr_2(k) is generated by the linear group \PGL(3,k) and the standard quadratic Cremona involution \sigma_2: [x:y:z] \mapsto [yz:xz:xy].^[65] This involution, a birational map of degree 2 with base points at [1:0:0], [0:1:0], and [0:0:1], was introduced by Luigi Cremona in his 1860s work on birational transformations of the plane.^[66]^[67] For \dim X \geq 2, \Bir(X) is infinite.^[68] Moreover, \Bir(X) acts naturally on the Picard group \Pic(X) by pulling back line bundles.^[69] In dimension 1, \Bir(\mathbb{P}^1_k) \cong \PGL(2,k), reflecting the birational equivalence of automorphisms of the projective line to projective linear transformations.^[70] The groups \Bir(\mathbb{P}^n_k) admit presentations in low dimensions: finitely presented in dimension 1, and presented via generators from \PGL(3,k) and \sigma_2 in dimension 2.^[70]^[71]

Structure and Classification

In dimension 2, the Cremona group Bir(\mathbb{P}^2_k) over an algebraically closed field k is generated by the projective linear group PGL(3,k) and quadratic transformations, such as the standard quadratic involution [X:Y:Z] \mapsto [YZ:XZ:XY].^[15] This finite generation result is given by the Noether-Castelnuovo theorem, which establishes that these elements suffice to produce all birational automorphisms of the projective plane.^[15] The structure of the birational automorphism group Bir(X) for a variety X incorporates the automorphism group Aut(X) as a subgroup, with Bir(X) acting on models obtained by point blow-ups; birational maps from X to such blow-ups \mathrm{Bl}_p X induce exact sequences relating Aut(\mathrm{Bl}_p X) to the kernel of the induced map on X.^[72] Classification of Bir(X) is achieved in specific cases, such as for abelian varieties where Bir(X) coincides with Aut(X), since any rational map to an abelian variety extends regularly.^[73] For rational surfaces, the structure aligns with that of the Cremona group, allowing explicit description via generators and relations, though Bir(X) exceeds Aut(X) in general.^[15] For varieties of general type, the birational automorphism group Bir(X) is finite.^[74] In higher dimensions, full classification remains open, with Bir(\mathbb{P}^n_k) for n \geq 3 lacking known minimal generating sets.^[15] Recent advances in the 2010s have employed Cox rings to classify varieties admitting torus actions, thereby elucidating the structure of their birational automorphism groups through graded ring automorphisms.^[75] The monoid of nef divisors further aids classification by parameterizing birational models via the nef cone, linking group actions to divisor theory on toric and Fano varieties.^[76] A key structural theorem states that Bir(\mathbb{P}^n) has virtual cohomological dimension n(n+1)/2.^[77]

Applications

In Complex and Algebraic Geometry

Birational geometry plays a pivotal role in the classification of complex and algebraic varieties by providing invariants that remain unchanged under birational transformations, enabling the study of equivalence classes of varieties sharing the same function field. In the context of compact complex surfaces, the Enriques-Kodaira classification theorem organizes minimal models into ten distinct classes based on birational invariants such as the Kodaira dimension, the second Betti number, and the canonical class. This classification, established through the analysis of birational contractions and resolutions, reveals that every compact complex surface is birationally equivalent to one of these minimal models, facilitating a complete birational taxonomy. The minimal model program (MMP), a cornerstone of birational geometry, has been instrumental in constructing moduli spaces for algebraic surfaces by producing good minimal models that parametrize families up to birational equivalence. For surfaces of general type, the birational MMP yields a projective moduli space that is locally isomorphic to the Hilbert scheme of canonical models, capturing the birational structure of the deformation space. In higher dimensions, progress remains partial, but the development of K-stability conditions in the 2020s has enabled the construction of moduli spaces for Fano varieties, where K-polystable Fanos form the boundary points of these compactifications, linking birational geometry to GIT quotients via test configurations.^[78]^[79]^[80] For hyperkähler manifolds, birational classification relies on Lagrangian fibrations, which are holomorphic maps to the base whose fibers are abelian varieties, providing a geometric tool to distinguish birational classes through the topology and period map of the base. These fibrations, when almost holomorphic, admit smooth good minimal models birational to the original manifold, allowing the study of birational transformations that preserve the hyperkähler structure and SYZ conjecture implications.^[81] In deformation theory, birational classes rigidify the possible deformations of complex varieties, as the Kodaira-Spencer cohomology and obstruction spaces are governed by birational invariants, ensuring that infinitesimal deformations remain within the same birational equivalence class for rigid varieties like those with ample canonical bundle.^[82] Recent advancements in the 2020s have filled key gaps in the MMP for Kähler pairs, extending the program from projective to non-projective settings by establishing termination of flips and abundance for log canonical pairs on Kähler threefolds, thereby providing a birational framework for classifying Kähler varieties beyond the algebraic realm.^[79]

In Arithmetic and Number Theory

Birational geometry plays a crucial role in the study of rational points on varieties over number fields, particularly through the invariance of certain cohomological groups under birational transformations. For a smooth projective variety X over a number field k, the group H^1(\mathrm{Gal}(\overline{k}/k), \mathrm{Pic}(\overline{X})) is a birational invariant, meaning that if Y is birationally equivalent to X, then H^1(\mathrm{Gal}(\overline{k}/k), \mathrm{Pic}(\overline{Y})) \cong H^1(\mathrm{Gal}(\overline{k}/k), \mathrm{Pic}(\overline{X})).^[83] This invariance facilitates the study of rational points via descent methods, where one reduces the problem of finding k-rational points on X to finding sections of torsors under the Picard group over the algebraic closure. Specifically, descent via models involves choosing a proper model of X over the ring of integers of k and using the birational equivalence to transfer the descent datum to a simpler model, often simplifying the computation of the Selmer group or the Tate-Shafarevich group associated to the Jacobian or Picard scheme.^[84] A prominent application arises in the Manin conjecture, which predicts the asymptotic distribution of rational points of bounded height on Fano varieties. The conjecture states that for a Fano variety X over a number field k with ample line bundle L = -K_X, the number of rational points x \in X(k) with height H_L(x) \leq B is asymptotically c_X B (\log B)^{r_X - 1} as B \to \infty, where c_X > 0 is a constant, and r_X = \mathrm{rk} \mathrm{Pic}(X) is the rank of the Picard group. Birational models are essential here, as the leading constants in the conjecture, including the self-intersection L^{\dim X} and the rank r_X, are birational invariants, allowing one to replace X by a birationally equivalent model (such as a toric variety or a blow-up) where the height counting is more tractable.^[85] For instance, on del Pezzo surfaces, birational transformations to weaker del Pezzo models preserve the asymptotic, enabling explicit computations and verifications of the conjecture in many cases.^[86] In the arithmetic of heights, birational invariance extends to pseudo-effective heights, which generalize the notion of height functions associated to big and nef divisors. For a projective variety X over a number field k, a height function h_D associated to a Cartier divisor D on X is pseudo-effective if it is non-negative on rational points and satisfies certain subadditivity properties, reflecting the arithmetic analog of pseudo-effective classes in the Néron-Severi group. Under birational maps \phi: X \dashrightarrow Y, such heights transform in a controlled way, preserving pseudo-effectivity because the pullback \phi^* D remains pseudo-effective if D is, allowing heights to be defined consistently across birational models. This invariance is crucial for applications in equidistribution theorems and Bogomolov-Miyaoka-Yau-type inequalities over number fields.^[87] A key theorem in this area asserts that birational maps preserve the Brauer-Manin obstruction to the existence of rational points. For smooth proper varieties X and Y over a number field k that are birationally equivalent via \phi: X \dashrightarrow Y, the Brauer group \mathrm{Br}(X) is isomorphic to \mathrm{Br}(Y), as both are isomorphic to the Brauer group of the function field k(X). Consequently, the Brauer-Manin pairing on X(k) \times \mathrm{Br}(X) corresponds to that on Y(k) \times \mathrm{Br}(Y), implying that if the Brauer-Manin obstruction obstructs rational points on X, it does so on Y, and vice versa. This result, due to the stable birational invariance of the Brauer group, has been instrumental in classifying varieties with no rational points, such as certain diagonal cubic surfaces.^[88] Recent developments include partial analogs of the minimal model program (MMP) in arithmetic settings over number fields, focusing on birational transformations that minimize arithmetic invariants like heights or discriminants. In the 2020s, progress has been made on establishing an arithmetic MMP for threefolds over rings of integers with residue characteristics greater than five, where one performs flips and contractions to obtain models with semi-ample canonical sheaf or bounded torsion index, adapting the complex MMP to control arithmetic obstructions like the Brauer group or class number. These partial results, while not yet complete for all dimensions or characteristics, provide tools for studying uniform boundedness of rational points on families of varieties.^[89]

References

[1]
[PDF] Birational geometry for number theorists - Brown Math
Birational geometry for number theorists explores the idea that geometry determines arithmetic, and provides background for conjectures of Bombieri, Lang and ...
[2]
[PDF] birational geometry of Deligne-Mumford stacks
Dec 21, 2023 · Introduction. Birational geometry of algebraic varieties is a classical subject in alge- braic geometry. Closely related to this is ...
[3]
Birational geometry of algebraic varieties, by János Kollár and ...
Feb 7, 2001 · This is based on the theory of extremal rays initiated by Mori and is central to the investigation of minimal models called the Minimal Model ...
[4]
Existence of minimal models for varieties of log general type - arXiv
Oct 5, 2006 · Existence of minimal models for varieties of log general type. Authors:Caucher Birkar, Paolo Cascini, Christopher D. Hacon, James McKernan.
[5]
[PDF] introduction to algebraic geometry, class 13
Definition. A rational map of prevarieties Y → X is the data of a map f : U →. X (where U is a non-empty hence dense open set of Y ) modulo the equivalence.
[6]
[PDF] Math 260X: Rationality Questions in Algebraic Geometry
Apr 6, 2022 · Definition 1.1. Let X and Y be varieties defined over a field k. 1. A rational map f : X 99K Y is an equivalence class of pairs (U, f) where U ...
[7]
[PDF] 15.1 Rational maps of affine varieties
Oct 29, 2013 · Before defining rational maps we want to nail down two points on which we we were inten- tional vague in the last lecture. We defined a morphism ...
[8]
[PDF] 1. Weil and Cartier divisors - KSU Math
So far we have established that any linear system with empty base locus defines a map from X to a projective space. This map is not always injective (see ...
[9]
Section 29.49 (01RR): Rational maps—The Stacks project
We say X and Y are birational if X and Y are isomorphic in the category of irreducible schemes and dominant rational maps. Assume X and Y are schemes over a ...
[10]
https://stacks.math.columbia.edu/tag/01RR
[11]
https://stacks.math.columbia.edu/tag/0BAC
[12]
TOWARDS BIRATIONAL CLASSIFICATION OF ALGEBRAIC ...
We say that two algebraic varieties are birationally equivalent if there is a birational map between them; this happens if and only if their function fields are ...
[13]
[PDF] 1. Succesive blow-ups for resolving the indeterminacy of - CUNY
Succesive blow-ups for resolving the indeterminacy of rational maps. Let ϕ : X 99K X0 be rational map between algebraic varieties. The.Missing: birational | Show results with:birational
[14]
[PDF] The Cremona group - Clay Mathematics Institute
The Cremona group Crn(k) is the group of k-automorphisms. of k(X1,...,Xn), the k-algebra of rational functions in n independent variables.
[15]
[PDF] Birational Geometry
Recall that two varieties X and Y over a field k are birational, if and only if there are non-empty open subsets U ⊂ X and V ⊂ Y that are isomorphic to each ...
[16]
[PDF] Birational classification of algebraic varieties - Berkeley Math
It is easy to see that two varieties are birational if they have the same field of rational functions C(X). ∼. = C(Y ). Recall that by Hironaka's theorem on ...
[17]
[1804.04078] Gabriel's theorem and birational geometry - arXiv
Apr 11, 2018 · This result interpolates between Gabriel's reconstruction theorem and the fact that two varieties are birational if and only if they have the ...Missing: iff | Show results with:iff
[18]
[PDF] RATIONALITY OF HYPERSURFACES 1. Introduction A variety X of ...
Oct 15, 2025 · birational geometry, Noether's problem, relations to the Cremona group, birational rigidity or derived category approaches to rationality ...
[19]
[PDF] plane conics in algebraic geometry - UChicago Math
Aug 29, 2014 · Q(X,Y,Z) = aX2 + bXY + cY 2 + dXZ + eY Z + fZ2, as q(x, y) = Q(X/Z ... All projective conics are of the form aX2 + cY 2 + fZ2 = 0 ...
[20]
(PDF) CONICS IN PROJECTIVE ALGEBRAIC GEOMETRY
Oct 27, 2025 · smooth conics correspond to a single moduli point, singular conics stratify the discriminant,. and the moduli perspective naturally accounts ...
[21]
[PDF] 3 Quadrics - People
Writing x, y as rational functions of t is why the process we have described is sometimes called the rational parametrization of the conic. It has its uses ...
[22]
[PDF] Biregular Cremona transformations of the plane - HAL
(2) Elements conjugate to a birational self-map of a conic bundle over P1 preserving the fiber ... First we identify C0 with P1 using the stereographic projection ...<|separator|>
[23]
[PDF] INTRODUCTION TO ALGEBRAIC GEOMETRY
Aug 27, 2020 · Let C be the conic in the projective plane P2 defined by f(x0 ... discriminant curve, and let Y be the double plane y2 = b2 − 4ac. (i) ...
[24]
[PDF] rational parametrization of conics
The set of K-rational points of L is denoted LK. Similarly, a conic curve defined over k is an equation. C : ax2 + bxy + cy2 + dx + ey + f = 0, a, b, c, d, e, f ...
[25]
[PDF] (1) The synthetic geometry of the Greeks, practically clo
Cayley added much to the theories of rational transformation and correspondence, showing the distinction between the theory of transformation of spaces and that ...
[26]
Geometry at Cambridge, 1863–1940 - ScienceDirect.com
This paper traces the ebbs and flows of the history of geometry at Cambridge from the time of Cayley to 1940, and therefore the arrival of a branch of modern ...
[27]
[PDF] Birational geometry of quadrics - UCLA Mathematics
Birational geometry of quadrics studies when two quadrics are birational, meaning they are birational over a field. Two quadric surfaces are birational if and ...
[28]
https://www.math.ucla.edu/~totaro/papers/public_html/birat.pdf
[29]
[PDF] 10. Algebraic Surfaces - Ziyu Zhang 张子宇
Proposition 10.7. A non-singular quadric surface is isomorphic to P1 × P1. Proof. We assume the quadric surface is S = V(z0z3 − z1z2).
[30]
[PDF] Blowing-up - Zürich - math.uzh.ch
Introduction Blowing-up is a fundamental concept of Algebraic Geometry and a subject which can be approached in quite different ways.<|control11|><|separator|>
[31]
Section 31.32 (01OF): Blowing up—The Stacks project
Blowing up is an important tool in algebraic geometry. ... The exceptional divisor of the blowup is the inverse image b^{-1}(Z). Sometimes Z is called the center ...
[32]
Resolution of Singularities - Purdue Math
In 1964, H.Hironaka published a celebrated paper and solved completely Resolution for any dimensional for characteristic zero. Grothendick once claimed ...
[33]
The Resolution of Singularities of an Algebraic Curve - jstor
ZARISKI. Introduction. The object of this n-ote is to give new anld brief proofs of the following well known theorems concerning the ...
[34]
[PDF] Resolution of Singularities: an Introduction - HAL
Theorem 1.6.1 (H. Hironaka [93] 1964) Every variety X over a ground field of characteristic zero admits a resolution of singularities.
[35]
[PDF] An informal introduction to blow-ups - Brown Math
In algebraic geometry, we make the question of replacing X by a manifold ˜X more interesting by requiring that the map ˜X → X be proper, which means it pulls ...Missing: paper | Show results with:paper
[36]
[PDF] notes for 483-3: kodaira dimension of algebraic varieties
It follows that h0(X,OX(C)) = 2, and so the linear system |C| is a pencil inducing a rational map f : X → P1. As C2 = 0, it follows (as in the proof of the ...
[37]
[PDF] arXiv:math/0205112v1 [math.AG] 10 May 2002
May 10, 2002 · singularity and the Euler characteristic of its projectivization. It ... exceptional divisor of the minimal embedded resolution of a plane curve.
[38]
[PDF] ON THE PROBLEM OF RESOLUTION OF SINGULARITIES IN ...
Jul 28, 2009 · The embedded resolution of singular algebraic varieties of dimension > 3 defined over fields of characteristic p > 0 is still an open problem.
[39]
[PDF] Artin Approximation - Journal of Singularities
The aim of this text is to present the Artin Approximation Theorem and some related results. The problem we are interested in is to find analytic solutions of ...
[40]
Complex Algebraic Surfaces
Arnaud Beauville, Université de Paris XI. Publisher: Cambridge ... PDF; Export citation. Select Chapter III - Ruled surfaces. Chapter III - Ruled ...
[41]
Compact Complex Surfaces - SpringerLink
In stockSeveral important developments have taken place in the theory of surfaces. The most sensational one concerns the differentiable structure of surfaces.
[42]
An Introduction to Birational Geometry of Algebraic Varieties
Free delivery 14-day returnsOct 14, 2011 · The aim of this book is to introduce the reader to the geometric theory of algebraic varieties, in particular to the birational geometry of algebraic varieties.
[43]
None
Summary of each segment:
[44]
3-FOLD LOG FLIPS - IOPscience
3-FOLD LOG FLIPS. V V Shokurov. © 1993 American Mathematical Society Izvestiya: Mathematics, Volume 40, Number 1Citation V V Shokurov 1993 Izv. Math. 40 95DOI ...Missing: paper | Show results with:paper
[45]
Termination of pseudo-effective 4-fold flips | Mathematische Zeitschrift
May 28, 2025 · As mentioned above, two of the main goals of the minimal model program are to prove the existence of flips and the termination of a sequence of ...
[46]
[PDF] 9. Birational invariants Definition 9.1. Let X be a normal projective ...
The Kodaira dimension of X is the Iitaka dimension of the canon- ical divisor, κ(X) = κ(X, KX). The plurigenera of X are the dimensions of the space of global.
[47]
[PDF] Chapter 1: Topology of algebraic varieties, Hodge decomposition ...
It is not true in general that all Hodge numbers are birational invariants. ... by various other methods); many references, especially of a more ...
[48]
[PDF] Birational Invariance of the S-fundamental Group Scheme
Feb 15, 2011 · It is well known that the etale fundamental group (or even topological or Nori fundamental group) of a smooth projective variety depends only on ...
[49]
[PDF] Hodge numbers of birational Calabi–Yau varieties via p-adic Hodge ...
Oct 6, 2023 · The theorems jointly prove that Hodge numbers are birational invariants for smooth projective Calabi–Yau varieties.
[50]
Uniruled and Rationally Connected Varieties - SpringerLink
A variety X is uniruled if there exists a rational curve through every point of X, whose deformations cover a dense open subset of X.
[51]
[PDF] How to classify Fano varieties?
For us, a Fano variety will be a smooth complex projective algebraic variety whose anticanonical bundle (i.e., the determinant of the tangent bundle) is ample.
[52]
[PDF] Fano varieties; Iskovskih's classification
Simplest examples are obtained by taking smooth complete intersections of type ... A basic invariant of a Fano manifold is its index: this is the maximal integer ...
[53]
rational connectedness and boundedness of fano manifolds - janos ...
JANOS KOLLAR, YOICHI MIYAOKA & SHIGEFUMI MORI is not always Fano. An example (a conic bundle) is found in [14]. This corollary was proved by [13] in a ...
[54]
PRAGMATIC 2010: Mori theory in a variety of flavors - mimuw
In the first of these papers, devoted to a proof of the Frankel-Hartshorne conjecture, Mori proved that Fano manifolds are covered by rational curves.Missing: theorem | Show results with:theorem
[55]
[PDF] Rigidity properties of Fano varieties - The Library at SLMath
By a recent result of Zhang [2006], it is known that Fano varieties are rationally connected ... Fano varieties with Picard number one”, J. Amer. Math. Soc ...
[56]
[PDF] rational curves and uniruled varieties
A rational curve is a curve C such that its normalization is P1, that is, a curve of geometric genus zero: g(C) = 0.
[57]
V. A. Iskovskikh, “Fano 3-folds. I”, Math. USSR-Izv., 11:3 (1977), 485 ...
Abstract: This article contains a classification of special Fano varieties; we give a description of the projective models of Fano 3-folds of index r⩾2 r ...
[58]
(PDF) G-Fano threefolds, I - ResearchGate
Aug 6, 2025 · We classify Fano threefolds with only terminal singularities whose canonical class is Cartier and divisible by 2, and satisfying an ...
[59]
[PDF] Boundedness results for singular Fano varieties, and applications to ...
This survey paper reports on work of Birkar, who confirmed a long-standing conjecture of Alexeev and Borisov-Borisov: Fano varieties with mild singularities ...Missing: BCHM | Show results with:BCHM
[60]
[1011.6131] Alpha invariant and K-stability of Q-Fano varieties - arXiv
Nov 29, 2010 · We give a purely algebro-geometric proof that if the alpha-invariant of a Q-Fano variety X is greater than dim X/(dim X+1), then (X,O(-K_X)) is K-stable.
[61]
[PDF] K-stability of Fano varieties - Math (Princeton)
The main goal of this book is to provide a comprehensive overview of the al- gebraic theory of K-stability for Fano varieties.
[62]
[PDF] arXiv:2208.07396v1 [math.AG] 15 Aug 2022
Aug 15, 2022 · The birational automorphism group of a variety X —denoted Bir(X )—is one of the most. natural birational invariants associated to X . For X = ℙ ...
[63]
[PDF] THE CREMONA GROUP AND ITS SUBGROUPS - HAL
Nov 12, 2020 · The third section deals with a geometric definition of birational maps of the complex pro- jective plane. 1.1. First definitions and examples.
[64]
[PDF] Relations in the Cremona group over a perfect field
The Cremona group Crn(k) = Birk(Pn) is the group of birational trans- formations of the projective n-space over a field k. In dimension n = 2 it has been ...
[65]
[PDF] arXiv:math/0408283v1 [math.AG] 20 Aug 2004
Aug 20, 2004 · We discuss the contribution of Luigi Cremona to the early development of the theory of cubic surfaces. 1. A brief history. In 1911 Archibald ...
[66]
[PDF] On the birational automorphism groups of algebraic varieties
It may actually occur that X and X' are birational while. Bir(X) and Bir(X') are not isomorphic. For example we let A be an abelian variety of dimension n ~ 2, ...
[67]
[PDF] arXiv:2111.09697v6 [math.AG] 24 Apr 2023
Apr 24, 2023 · Let X be a surface and G be an algebraic subgroup of Bir(X) such that G◦ acts regularly on X. ... (X) acts non-trivially on Pic(X) by.
[68]
[PDF] Representations of finite subgroups of Cremona groups - arXiv
Jul 8, 2025 · The line Cremona group Cr1(k) is isomorphic to the linear algebraic group. PGL2(k). Even in this case, the representation theory of its finite ...
[69]
[PDF] A NEW PRESENTATION OF THE PLANE CREMONA GROUP
We give a presentation of the plane Cremona group over an alge- braically closed field with respect to the generators given by the Theorem of. Noether and ...
[70]
[PDF] Birational automorphisms of varieties
May 11, 2023 · Any embedding of a finite group G to GL2(C) induces a faithful 2-dimensional representation of the group G. The dimension of any irreducible ...
[71]
When does Aut(X)=Bir(X) hold? - MathOverflow
Jan 31, 2013 · If X is smooth projective and contains no rational curves, then its automorphism group is equal to the group of its birational endomorphisms.Birational Automorphisms and infinite divisibility - MathOverflowReference request: birational automorphism group is finiteMore results from mathoverflow.netMissing: exact sequence Bl_p
[72]
Cox rings, semigroups and automorphisms of affine algebraic varieties
Oct 7, 2008 · We show that each automorphism of an affine variety can be lifted to an automorphism of the Cox ring normalizing the grading. It follows that ...Missing: birational | Show results with:birational
[73]
Cox Rings - Cambridge University Press & Assessment
Automorphism groups of generic rational surfaces. J. Algebra, 116(1):130–142 ... Algebraic groups and their birational invariants. Translations of ...
[74]
[1210.6960] Topologies and structures of the Cremona groups - arXiv
Oct 25, 2012 · We study the algebraic structure of the n-dimensional Cremona group and show that it is not an algebraic group of infinite dimension (ind-group) ...Missing: virtual cohomological
[75]
New directions in the Minimal Model Program
Sep 12, 2020 · Such a vertex Y is called the minimal model of X and, by Castelnuovo's theorem, it is characterised by the fact that it does not admit any ...
[76]
The log minimal model program for Kähler $3$-folds - arXiv
Sep 13, 2020 · In this article we show that the Log Minimal Model Program for \mathbb{Q}-factorial dlt pairs (X, B) on a compact Kähler 3-fold holds.
[77]
On properness of K-moduli spaces and optimal degenerations of ...
We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on ...
[78]
[1105.3410] Lagrangian fibrations on hyperkähler manifolds - arXiv
May 17, 2011 · Beauville posed the question whether X admits a Lagrangian fibration with fibre L. We show that this is indeed the case if X is not projective.
[79]
[PDF] Bimeromorphic geometry of Kähler threefolds - Université Côte d'Azur
We describe the recently established minimal model program for. (non-algebraic) Kähler threefolds as well as the abundance theorem for these spaces. 1.
[80]
[PDF] Birational geometry of algebraic tori
Then the Π-modules Pic ¯X and Pic¯Y are similar and groups H1(Π, Pic ¯X) and. H−1(Π, Pic ¯X) are birational invariants of X. Let G be a connected linear ...<|control11|><|separator|>
[81]
[PDF] Rational points on varieties - MIT Mathematics
Jun 6, 2010 · ... rational map, dominant. Definition 2.0.1. If S is a scheme, an S ... projective varieties Mg,n and M. (L) g,n. 2.3.4. Functorial ...
[82]
On upper bounds of Manin type - Project Euclid
We introduce a certain birational invariant of a polarized algebraic variety and use that to obtain upper bounds for the counting functions of rational ...
[83]
[PDF] on exceptional sets in manin's conjecture - Sites@BC
The simplest and best solution is to use birational invariance: we define a(X, L) by taking any resolution of singularities φ : X0 → X and setting a(X, L) := a( ...
[84]
[PDF] campana points, height zeta functions, - and log manin's conjecture
Now we are ready to introduce two birational invariants which play central roles in Manin's conjecture: Definition 4.1. Let (X, De) be a klt Campana ...
[85]
[PDF] Rational points on varieties and the Brauer-Manin obstruction
(4) The Brauer group is a birational invariant of smooth projective varieties, i.e., if f: X 99K Y is a birational map between two smooth projective varieties, ...
[86]
regular varieties and the minimal model program for threefolds in ...
Dec 31, 2020 · This paper establishes the Minimal Model Program for arithmetic threefolds with residue characteristics greater than five, generalizing global ...