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Field of algebraic geometry
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside
Birational_geometry
canonical ring and therefore likewise the Kodaira dimension is a birational invariant: Any birational map between smooth compact complex manifolds induces an isomorphism
Canonical_ring
Algebraic variety of dimension two
fundamental theorems for the birational geometry of surfaces is Castelnuovo's theorem. This states that any birational map between algebraic surfaces
Algebraic_surface
Property in algebraic geometry
In algebraic geometry, a birational invariant is a property that is preserved under birational equivalence. A birational invariant is a quantity or object
Birational_invariant
Chinese mathematician
in the area of algebraic geometry and a professor at Princeton University. Xu is known for his work in birational geometry, the minimal model program
Chenyang_Xu
Group of Italian mathematicians who studied birational geometry (c. 1885–1935)
mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered
Italian school of algebraic geometry
Italian_school_of_algebraic_geometry
Effort to birationally classify algebraic varieties
algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model
Minimal_model_program
Concept in algebraic geometry
1965. Iitaka 1970. Iitaka 1971. J. A. Chen and M. Chen, Explicit birational geometry of 3-folds and 4-folds of general type III, Theorem 1.4. O. Fujino
Kodaira_dimension
Geometric space
In algebraic geometry, a moduli space of curves is a space whose points correspond to isomorphism classes of algebraic curves. The term "modulus" was introduced
Moduli_of_algebraic_curves
Branch of mathematics
giving the fundamental Kleinian geometry on projective space, they concerned themselves also with the higher-degree birational transformations. This weaker
Algebraic_geometry
Kurdish mathematician
modern birational geometry. In 2010 he received the Leverhulme Prize in mathematics and statistics for his contributions to algebraic geometry, and in
Caucher_Birkar
Italian mathematician (1871–1946)
a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry. Enriques was born in Livorno, and brought
Federigo_Enriques
Type of geometric transformation
Blowups are the most fundamental transformation in birational geometry, because every birational morphism between projective varieties is a blowup. The
Blowing_up
Surgery operation in minimal model program
dimension 3 flips are used to construct minimal models, and any two birationally equivalent minimal models are connected by a sequence of flops. It is
Flip_(algebraic_geometry)
Surface containing a line through every point
In geometry, a surface S in 3-dimensional Euclidean space is ruled (also called a scroll) if through every point of S, there is a straight line that lies
Ruled_surface
Russian mathematician (born 1950)
awarded his Ph.D. ("candidate degree") in 1976. Shokurov works on the birational geometry of algebraic varieties. After obtaining his Ph.D., he worked at the
Vyacheslav_Shokurov
Mathematical algorithm
In arithmetic geometry, the Cox–Zucker machine is an algorithm introduced by David A. Cox and Steven Zucker for studying elliptic surfaces. It determines
Cox–Zucker_machine
Kind of partial function between algebraic varieties
tracing through the proof of the theorem it is possible to do so. Birational geometry Blowing up Function field of an algebraic variety Resolution of singularities
Rational_mapping
Study of complex manifolds and several complex variables
tremendously both from techniques in analysis and in pure birational geometry. Complex geometry has significant applications to theoretical physics, where
Complex_geometry
Group in birational geometry
In birational geometry, the Cremona group, named after Luigi Cremona, is the group of birational automorphisms of the n {\displaystyle n} -dimensional
Cremona_group
Swiss mathematician
of Basel. Her research focuses on algebraic geometry, and especially Cremona groups in birational geometry. Zimmermann is from Glarus Süd; she was born
Susanna_Zimmermann
Curve from a cone intersecting a plane
type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is
Conic_section
Concept in algebraic geometry
{\displaystyle X} is a combinatorial invariant of importance to the birational geometry of X {\displaystyle X} . Let X {\displaystyle X} be a proper variety
Cone_of_curves
Mathematical concept
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic
Elliptic_surface
Overview of and topical guide to geometry
Absolute geometry Affine geometry Algebraic geometry Analytic geometry Birational geometry Complex geometry Computational geometry Conformal geometry Constructive
Outline_of_geometry
Curve defined as zeros of polynomials
Hilbert's sixteenth problem Cubic plane curve Hyperelliptic curve Birational geometry Conic section Elliptic curve Fractional ideal Function field of an
Algebraic_curve
crucial in birational geometry: an elementary result (see for instance Shafarevich, II.4.4) shows (under suitable assumptions) that any birational regular
Exceptional_divisor
In algebraic geometry, the abundance conjecture is a conjecture in birational geometry, more precisely in the minimal model program, stating that for
Abundance_conjecture
to the general n. Therefore The study of the higher-dimensional birational geometry decompose to the part of κ=-∞,0,n and the fiber space whose fibers
Iitaka_dimension
2-dimensional complex projective space
the 5-sphere, i.e. torsion. In birational geometry, a complex rational surface is any algebraic surface birationally equivalent to the complex projective
Complex_projective_plane
Romanian-American mathematician
Harvard University, specializing in algebraic geometry. He is known for his work on complex birational geometry, Hodge theory, abelian varieties, and vector
Mihnea_Popa
Mathematical classification of surfaces
P_{n}=\dim H^{0}(K^{n}),n\geqslant 1} are called the plurigenera. They are birational invariants, i.e., invariant under blowing up. Using Seiberg–Witten theory
Enriques–Kodaira classification
Enriques–Kodaira_classification
Theorem in algebraic geometry
Theorem in algebraic geometry
Lüroth's_theorem
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
surfaces and Mori fiber spaces. The following perspective is crucial in birational geometry (in particular in Mori's minimal model program). Let X {\displaystyle
Contraction_morphism
Russian mathematician (1939–2009)
seminar and, following the recommendation of Yuri Manin, studied birational geometry. After graduating in 1964, Iskovskikh entered the graduate school
Vasilii_Iskovskikh
Brazilian mathematician
is a Brazilian mathematician specializing in algebraic geometry, including birational geometry, Fano varieties, and foliations. Other than her research
Carolina Araujo (mathematician)
Carolina_Araujo_(mathematician)
The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Concept in algebraic geometry
In algebraic geometry, an infinitely near point of an algebraic surface S is a point on a surface obtained from S by repeatedly blowing up points. Infinitely
Infinitely_near_point
Hungarian mathematician
OCLC 33243194. Kollár, János; Mori, Shigefumi; Clemens, C. Herbert (1998). Birational geometry of algebraic varieties. Cambridge: Cambridge University Press. ISBN 0-521-63277-3
János_Kollár
Scheme theory concept
geometry can exhibit discontinuities of a kind that are detected by flatness. For instance, the operation of blowing down in the birational geometry of
Flat_morphism
Geometric theorem
In geometry, Hesse's principle of transfer (German: Übertragungsprinzip) states that if the points of the projective line P1 are depicted by a rational
Hesse's_principle_of_transfer
Notion in algebraic geometry
proved to be very useful in various branches of algebraic geometry, most notably birational geometry and singularity theory. Roughly speaking, motivic integration
Motivic_integration
Concept in algebraic geometry
linear systems became a basic tool of birational geometry as practised by the Italian school of algebraic geometry. The technical demands became quite stringent;
Linear_system_of_divisors
Algebraic surface with special triviality properties
In mathematics, Enriques surfaces are algebraic surfaces such that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial
Enriques_surface
Algebraic variety
rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means
Rational_variety
Generalization of algebraic variety
morphism, Proper morphism, Finite morphism, Étale morphism Stable curve Birational geometry Étale cohomology, Chow group, Hodge theory Group scheme, Abelian
Scheme_(mathematics)
In algebraic geometry, a variety over a field k {\displaystyle k} is ruled if it is birational to the product of the projective line with some variety
Ruled_variety
Algebraic variety that is a moduli space for principally polarized abelian varieties
Klaus; Sankaran, G. K. (2002). "The Geometry of Siegel Modular Varieties". Higher Dimensional Birational Geometry. Advanced Studies in Pure Mathematics
Siegel_modular_variety
German mathematician (1844–1921)
given degree d from an algebraic curve to projective space Pn. In birational geometry, Noether introduced the fundamental technique of blowing up in order
Max_Noether
Completion of the usual space with "points at infinity"
space. In this case one obtains a so-called rational map, see also Birational geometry.) Two linear maps S and T in L(V, W) induce the same map between
Projective_space
Surface in algebraic geometry
In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational
Rational_surface
singularity Crepant resolution Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134
Discrepancy (algebraic geometry)
Discrepancy_(algebraic_geometry)
Ample line bundle Ample vector bundle Linear system of divisors Birational geometry Blowing up Resolution of singularities Rational variety Unirational
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Line with a point at infinity added
has a subfield isomorphic with K(T). From the point of view of birational geometry, this means that there will be a rational map from V to P1(K), that
Projective_line
Theory in number theory
2022-01-31. Pop, Florian (1994), "On Grothendieck's conjecture of birational anabelian geometry", Annals of Mathematics, (2), 139 (1): 145–182, doi:10.2307/2946630
Anabelian_geometry
French mathematician (1928–2014)
expressiveness as well as its technical depth. In that setting one can use birational geometry, techniques from number theory, Galois theory, commutative algebra
Alexander_Grothendieck
JSTOR 2373050, MR 0199191 Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134
Rational_singularity
Singularities of algebraic varieties
ISBN 978-1-107-03534-8, MR 3057950 Kollár, János; Mori, Shigefumi (1998), Birational Geometry of Algebraic Varieties, Cambridge University Press, doi:10.1017/CBO9780511662560
Canonical_singularity
a wide range of topics in biology. Birational geometry a part of algebraic geometry that deals with the geometry (of an algebraic variety) that is dependent
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Structure in algebraic geometry
relaxed question of studying varieties up to birational isomorphism has led to the field of birational geometry. Another way to handle the question is to
Motive_(algebraic_geometry)
Topics referred to by the same term
papyrus Matrilysin, an enzyme Minimal model program, a branch of birational geometry Million progressive motile (million motile sperm cells per milliliter)
MMP
Concept in algebraic geometry
one of the most powerful formulas in algebraic geometry. An important tool of modern birational geometry is inversion of adjunction, which allows one to
Canonical_bundle
Branch of algebraic geometry
exceptional curve of a blow-up, which is a central operation in birational geometry. Given an algebraic surface S, blowing up at a point creates a curve
Intersection_theory
Mathematical theory
Hironaka's fundamental theorem on resolution of singularities (in birational geometry in characteristic 0). This means that the simple process of "lifting"
Singularity_theory
Mathematical object studied in the field of algebraic geometry
mathematical meanings Function field of an algebraic variety Birational geometry Motive (algebraic geometry) Analytic variety Zariski–Riemann space Semi-algebraic
Algebraic_variety
Mathematics award
rigidity of sol" "For their work in advancing our understanding of the birational geometry of algebraic varieties in dimension greater than three, in particular
Clay_Research_Award
powerful alternative characterisations in terms of the algebraic and birational geometry of the Fano variety. Thus in the case of Fano varieties, there are
K-stability_of_Fano_varieties
In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold
Surface_of_general_type
British mathematician
Samuel, P. (1955). "Review: Methods of algebraic geometry. Vol. III. Birational geometry. By W. V. D. Hodge and D. Pedoe" (PDF). Bull. Amer. Math
W._V._D._Hodge
Generalizations of codimension-1 subvarieties of algebraic varieties
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common
Divisor_(algebraic_geometry)
Surface in algebraic geometry
degree n in projective space of dimension n + 1. Here "rational" means birational to projective space, "scroll" is an old term for ruled surface, and "normal"
Rational_normal_scroll
Mathematical term; concerning axioms used to derive theorems
included non-Euclidean geometry, Georg Cantor's abstract set theory, and Hilbert's revisionist axioms for Euclidean geometry. David Hilbert "was the
Axiomatic_system
Type of commutative ring in mathematics
ISBN 978-3-540-62046-4, MR 1644323 Kollár, János; Mori, Shigefumi (1998), Birational Geometry of Algebraic Varieties, Cambridge University Press, doi:10.1017/CBO9780511662560
Cohen–Macaulay_ring
Algebraic structure with addition, multiplication, and division
and their geometric meaning in higher dimensions is referred to as birational geometry. The minimal model program attempts to identify the simplest (in
Field_(mathematics)
Italian mathematician (1879–1961)
the French Academy of Sciences. He contributed in a major way to birational geometry, the theory of algebraic surfaces, in particular of the curves lying
Francesco_Severi
Branch of mathematics
{\displaystyle 3g-3} . Deformation theory was famously applied in birational geometry by Shigefumi Mori to study the existence of rational curves on varieties
Deformation_(mathematics)
Algebraic curve
model (also called a smooth completion), equivalent in the sense of birational geometry, is meant. To be more precise, the equation defines a quadratic extension
Hyperelliptic_curve
In algebraic geometry, the smooth completion (or smooth compactification) of a smooth affine algebraic curve X is a complete smooth algebraic curve which
Smooth_completion
Property of algebraic varieties and complex manifolds
In algebraic geometry, the geometric genus is a basic birational invariant pg of algebraic varieties and complex manifolds. The geometric genus can be
Geometric_genus
Concept in algebraic geometry
the quotient variety. Unfortunately this — the point of view of birational geometry — can only give a first approximation to the answer. As Mumford put
Geometric_invariant_theory
Japanese mathematician (born 1951)
JSTOR 2951828. S2CID 17830187. Kollár, János; Mori, Shigefumi. Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and
Shigefumi_Mori
Japanese mathematician (1915–1997)
the classification of algebraic surfaces from the point of view of birational geometry of complex manifolds. This resulted in a typology of seven kinds
Kunihiko_Kodaira
Romanian-American mathematician
singularities, jet schemes, D-modules or positive characteristic methods ... birational geometry, asymptotic base loci and invariants of divisors, and toric varieties
Mircea_Mustață
Study of systems of inequalitites
In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations
Real_algebraic_geometry
Concept in algebraic geometry
(PDF), Journal of Algebraic Geometry, 3: 295–345, MR 1257325 Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge
Nef_line_bundle
Space of complex matrices with positive definite imaginary part
Klaus; Sankaran, G. K. (2002). "The geometry of Siegel modular varieties". Higher Dimensional Birational Geometry (Kyoto, 1997). Advanced Studies in Pure
Siegel_upper_half-space
is a rational normal curve. Rational normal scroll Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York. ISBN 0-387-97716-3
Rational_normal_curve
properties of X which depend only on KX. This is the subject of birational geometry. If X is an algebraic variety over a field k, then over each open
Function field (scheme theory)
Function_field_(scheme_theory)
Russian-American mathematician (1899–1986)
seal of Zariski's discontent with the approach of the Italians to birational geometry. He addressed the question of rigour by recourse to commutative algebra
Oscar_Zariski
Concept in algebraic geometry
{\text{rank}}(F)} . A useful weakening of ampleness, notably in birational geometry, is the notion of a big line bundle. A line bundle L on a projective
Ample_line_bundle
Mathematical concept in algebraic geometry
variety V that depend only on the function field are studied in birational geometry. The function field of a point over K is K. The function field of
Function field of an algebraic variety
Function_field_of_an_algebraic_variety
Topics referred to by the same term
refer to: Minimal model (birational geometry), classification of algebraic varieties with the goal to construct a birational model of any complex projective
Minimal_model
Series of mathematics textbooks
Hochschild (1981, ISBN 978-1-4613-8116-7) Algebraic Geometry – An Introduction to Birational Geometry of Algebraic Varieties, Shigeru Iitaka (1982,
Graduate_Texts_in_Mathematics
Theorem of algebraic geometry and commutative algebra
In algebraic geometry, Zariski's main theorem, proved by Oscar Zariski (1943), is a statement about the structure of birational morphisms stating roughly
Zariski's_main_theorem
Israeli-American mathematician
Professor at Brown University. Among other topics, he has dealt with birational geometry, the resolution of singularities, subvarieties of abelian varieties
Dan_Abramovich
Algebraic geometry scheme
ISBN 978-1-107-03534-8, MR 3057950 Kollár, János; Mori, Shigefumi (1998), Birational Geometry of Algebraic Varieties, Cambridge University Press, ISBN 0-521-63277-3
Gorenstein_scheme
H 2 {\displaystyle \mathbb {H} ^{2}} , each direct central conic is birationally equivalent to an opposite central conic. In fact, the central conics
Steiner_conic
Algebraic structure with addition and multiplication
algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies
Ring_(mathematics)
English-born mathematician and geometer
Samuel, P. (1955). "Review: Methods of algebraic geometry Vol. 3. Birational geometry by W. V. D. Hodge and D. Pedoe". Bull. Amer. Math. Soc. 61
Daniel_Pedoe
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