Category: Theorems in algebraic geometry

Appell–Humbert theorem
In mathematics, the Appell–Humbert theorem describes the line bundles on a complex torus or complex abelian variety.It was proved for 2-dimensional tori by Appell and Humbert, and in general by Lefsch
Borel's theorem
In topology, a branch of mathematics, Borel's theorem, due to Armand Borel, says the cohomology ring of a classifying space or a classifying stack is a polynomial ring.
Harnack's curve theorem
In real algebraic geometry, Harnack's curve theorem, named after Axel Harnack, gives the possible numbers of connected components that an algebraic curve can have, in terms of the degree of the curve.
Grothendieck trace formula
In algebraic geometry, the Grothendieck trace formula expresses the number of points of a variety over a finite field in terms of the trace of the Frobenius endomorphism on its cohomology groups. Ther
Weber's theorem
In mathematics, Weber's theorem, named after Heinrich Martin Weber, is a result on algebraic curves. It states the following. Consider two non-singular curves C and C′ having the same genus g > 1. If
Hirzebruch–Riemann–Roch theorem
In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on
Abhyankar–Moh theorem
In mathematics, the Abhyankar–Moh theorem states that if is a complex line in the complex affine plane , then every embedding of into extends to an automorphism of the plane. It is named after Shreera
Tate's isogeny theorem
In mathematics, Tate's isogeny theorem, proved by Tate, states that two abelian varieties over a finite field are isogeneous if and only if their Tate modules are isomorphic (as Galois representations
Zariski's main theorem
In algebraic geometry, Zariski's main theorem, proved by Oscar Zariski, is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of
Raynaud's isogeny theorem
In mathematics, Raynaud's isogeny theorem, proved by Raynaud, relates the Faltings heights of two isogeneous elliptic curves.
Theorem on formal functions
In algebraic geometry, the theorem on formal functions states the following: Let be a proper morphism of noetherian schemes with a coherent sheaf on X. Let be a closed subscheme of S defined by and fo
Behrend's trace formula
In algebraic geometry, Behrend's trace formula is a generalization of the Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite field, conjectured in 1993 and proven in 2003 b
Mumford vanishing theorem
In algebraic geometry, the Mumford vanishing theorem proved by Mumford in 1967 states that if L is a semi-ample invertible sheaf with Iitaka dimension at least 2 on a complex projective manifold, then
Beauville–Laszlo theorem
In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebra
Krivine–Stengle Positivstellensatz
In real algebraic geometry, Krivine–Stengle Positivstellensatz (German for "positive-locus-theorem") characterizes polynomials that are positive on a semialgebraic set, which is defined by systems of
Grothendieck's connectedness theorem
In mathematics, Grothendieck's connectedness theorem , states that if A is a complete Noetherian local ring whose spectrum is k-connected and f is in the maximal ideal, then Spec(A/fA) is (k − 1)-conn
Schlessinger's theorem
In algebra, Schlessinger's theorem is a theorem in deformation theory introduced by Schlessinger that gives conditions for a functor of artinian local rings to be pro-representable, refining an earlie
Lang's theorem
In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if G is a connected smooth algebraic group over a finite field , then, writing for the Frobenius, the morphism of varieties is
Sumihiro's theorem
In algebraic geometry, Sumihiro's theorem, introduced by, states that a normal algebraic variety with an action of a torus can be covered by torus-invariant affine open subsets. The "normality" in the
Weil reciprocity law
In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field K(C) of an algebraic curve C over an algebraically closed field K. Given functions f and g in K(C), i.e
Nakano vanishing theorem
In mathematics, specifically in the study of vector bundles over complex Kähler manifolds, the Nakano vanishing theorem, sometimes called the Akizuki–Nakano vanishing theorem, generalizes the Kodaira
Enriques–Babbage theorem
In algebraic geometry, the Enriques–Babbage theorem states that a canonical curve is either a set-theoretic intersection of quadrics, or trigonal, or a plane quintic. It was proved by Babbage and Enri
Riemann–Roch theorem for surfaces
In mathematics, the Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. The classical form of it was first given by Castelnuovo , after preliminary ver
Reiss relation
In algebraic geometry, the Reiss relation, introduced by Reiss, is a condition on the second-order elements of the points of a plane algebraic curve meeting a given line.
Gram's theorem
In mathematics, Gram's theorem states that an algebraic set in a finite-dimensional vector space invariant under some linear group can be defined by absolute invariants. (Dieudonné & Carrell , p. 31).
Torelli theorem
In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (co
Abhyankar's lemma
In mathematics, Abhyankar's lemma (named after Shreeram Shankar Abhyankar) allows one to kill tame ramification by taking an extension of a base field. More precisely, Abhyankar's lemma states that if
Porteous formula
In mathematics, the Porteous formula, or Thom–Porteous formula, or Giambelli–Thom–Porteous formula, is an expression for the fundamental class of a degeneracy locus (or determinantal variety) of a mor
ELSV formula
In mathematics, the ELSV formula, named after its four authors , , , , is an equality between a Hurwitz number (counting ramified coverings of the sphere) and an integral over the moduli space of stab
Theorem of Bertini
In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by E
Chow's lemma
Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry. It roughly says that a proper morphism is fairly close to being a projective morphism. More precisel
Cartan's theorems A and B
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X. They are significant both as applied to several compl
Honda–Tate theorem
In mathematics, the Honda–Tate theorem classifies abelian varieties over finite fields up to isogeny. It states that the isogeny classes of simple abelian varieties over a finite field of order q corr
Lange's conjecture
In algebraic geometry, Lange's conjecture is a theorem about stability of vector bundles over curves, introduced by and proved by Montserrat Teixidor i Bigas and in 1999.
Kempf–Ness theorem
In algebraic geometry, the Kempf–Ness theorem, introduced by George Kempf and Linda Ness, gives a criterion for the stability of a vector in a representation of a complex reductive group. If the compl
Borel fixed-point theorem
In mathematics, the Borel fixed-point theorem is a fixed-point theorem in algebraic geometry generalizing the Lie–Kolchin theorem. The result was proved by Armand Borel.
Tsen's theorem
In mathematics, Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1). This implies that the Brauer group of a
Regular embedding
In algebraic geometry, a closed immersion of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of is generated by a regula
Ribet's theorem
Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jea
Lefschetz hyperplane theorem
In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and th
Veblen–Young theorem
In mathematics, the Veblen–Young theorem, proved by Oswald Veblen and John Wesley Young , states that a projective space of dimension at least 3 can be constructed as the projective space associated t
Tarski–Seidenberg theorem
In mathematics, the Tarski–Seidenberg theorem states that a set in (n + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto n-dimensional space, and the re
Grauert–Riemenschneider vanishing theorem
In mathematics, the Grauert–Riemenschneider vanishing theorem is an extension of the Kodaira vanishing theorem on the vanishing of higher cohomology groups of coherent sheaves on a compact complex man
Bézout's theorem
Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of n polynomials in n indeterminates. In its original form the theorem states that in general the number of
Lefschetz theorem on (1,1)-classes
In algebraic geometry, a branch of mathematics, the Lefschetz theorem on (1,1)-classes, named after Solomon Lefschetz, is a classical statement relating holomorphic line bundles on a compact Kähler ma
Zariski's connectedness theorem
In algebraic geometry, Zariski's connectedness theorem (due to Oscar Zariski) says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's ma
Luna's slice theorem
In mathematics, Luna's slice theorem, introduced by , describes the local behavior of an action of a reductive algebraic group on an affine variety. It is an analogue in algebraic geometry of the theo
AF+BG theorem
In algebraic geometry the AF+BG theorem (also known as Max Noether's fundamental theorem) is a result of Max Noether that asserts that, if the equation of an algebraic curve in the complex projective
Kawasaki's Riemann–Roch formula
In differential geometry, Kawasaki's Riemann–Roch formula, introduced by Tetsuro Kawasaki, is the Riemann–Roch formula for orbifolds. It can compute the Euler characteristic of an orbifold. Kawasaki's
Cayley–Bacharach theorem
In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane P2. The original form states: Assume that two cubics C1 and C2 in
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra
Leray's theorem
In algebraic topology and algebraic geometry, Leray's theorem (so named after Jean Leray) relates abstract sheaf cohomology with Čech cohomology. Let be a sheaf on a topological space and an open cove
Serre's theorem on affineness
In the mathematical discipline of algebraic geometry, Serre's theorem on affineness (also called Serre's cohomological characterization of affineness or Serre's criterion on affineness) is a theorem d
Chevalley's structure theorem
In algebraic geometry, Chevalley's structure theorem states that a smooth connected algebraic group over a perfect field has a unique normal smooth connected affine algebraic subgroup such that the qu
Ramanujam–Samuel theorem
In algebraic geometry, the Ramanujam–Samuel theorem gives conditions for a divisor of a local ring to be principal. It was introduced independently by Samuel in answer to a question of Grothendieck an
Chasles' theorem (geometry)
In algebraic geometry, Chasles' theorem says that if two pencils of curves have no curves in common, then the intersections of those curves form another pencil of curves the degree of which can be cal
Kodaira vanishing theorem
In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indice
Chevalley–Iwahori–Nagata theorem
In mathematics, the Chevalley–Iwahori–Nagata theorem states that if a linear algebraic group G is acting linearly on a finite-dimensional vector space V, then the map from V/G to the spectrum of the r
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional
Kodaira embedding theorem
In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex man
Serre–Tate theorem
In algebraic geometry, the Serre–Tate theorem says that an abelian scheme and its p-divisible group have the same infinitesimal deformation theory. This was first proved by Jean-Pierre Serre when the
Gabriel–Rosenberg reconstruction theorem
In algebraic geometry, the Gabriel–Rosenberg reconstruction theorem, introduced in , states that a quasi-separated scheme can be recovered from the category of quasi-coherent sheaves on it. The theore
Reider's theorem
In algebraic geometry, Reider's theorem gives conditions for a line bundle on a projective surface to be very ample.
Hodge index theorem
In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly speaking, that the space spanne
Keel–Mori theorem
In algebraic geometry, the Keel–Mori theorem gives conditions for the existence of the quotient of an algebraic space by a group. The theorem was proved by Sean Keel and Shigefumi Mori. A consequence
Grothendieck–Riemann–Roch theorem
In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem
Gudkov's conjecture
In real algebraic geometry, Gudkov's conjecture, also called Gudkov’s congruence, (named after Dmitry Gudkov) was a conjecture, and is now a theorem, which states that an M-curve of even degree obeys
Hurwitz's automorphisms theorem
In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that
Lüroth's theorem
In mathematics, Lüroth's theorem asserts that every field that lies between two other fields K and K(X) must be generated as an extension of K by a single element of K(X). This result is named after J
Holomorphic Lefschetz fixed-point formula
In mathematics, the Holomorphic Lefschetz formula is an analogue for complex manifolds of the Lefschetz fixed-point formula that relates a sum over the fixed points of a holomorphic vector field of a
Mnëv's universality theorem
In algebraic geometry, Mnëv's universality theorem is a result which can be used to represent algebraic (or semi algebraic) varieties as realizations of oriented matroids, a notion of combinatorics.
Birkhoff–Grothendieck theorem
In mathematics, the Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over is a direct sum of holomorp
De Franchis theorem
In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1
Atiyah–Bott formula
In algebraic geometry, the Atiyah–Bott formula says the cohomology ring of the moduli stack of principal bundles is a free graded-commutative algebra on certain homogeneous generators. The original wo
Base change theorems
In mathematics, the base change theorems relate the direct image and the inverse image of sheaves. More precisely, they are about the base change map, given by the following natural transformation of
Nagata's compactification theorem
In algebraic geometry, Nagata's compactification theorem, introduced by Nagata , implies that every abstract variety can be embedded in a complete variety, and more generally shows that a separated an
Grothendieck existence theorem
In mathematics, the Grothendieck existence theorem, introduced by Grothendieck , gives conditions that enable one to lift infinitesimal deformations of a scheme to a deformation, and to lift schemes o
Kempf vanishing theorem
In algebraic geometry, the Kempf vanishing theorem, introduced by Kempf, states that the higher cohomology group Hi(G/B,L(λ)) (i > 0) vanishes whenever λ is a dominant weight of B. Here G is a reducti
Néron–Ogg–Shafarevich criterion
In mathematics, the Néron–Ogg–Shafarevich criterion states that if A is an elliptic curve or abelian variety over a local field K and ℓ is a prime not dividing the characteristic of the residue field
Projection formula
In algebraic geometry, the projection formula states the following: For a morphism of ringed spaces, an -module and a locally free -module of finite rank, the natural maps of sheaves are isomorphisms.
Torsion conjecture
In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian varieties states that the order of the torsion group of an abelian vari
Belyi's theorem
In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified c
Noether's theorem on rationality for surfaces
In mathematics, Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, giving a criterion for a rational surface. Let S be an algebraic surfa
Faltings's theorem
In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points.
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions wit
Fulton–Hansen connectedness theorem
In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to ma
Kawamata–Viehweg vanishing theorem
In algebraic geometry, the Kawamata–Viehweg vanishing theorem is an extension of the Kodaira vanishing theorem, on the vanishing of coherent cohomology groups, to logarithmic pairs, proved independent
Chasles–Cayley–Brill formula
In algebraic geometry, the Chasles–Cayley–Brill formula, also known as the Cayley–Brill formula, states that a correspondence T of valence k from an algebraic curve C of genus g to itself has d + e +
Clifford's theorem on special divisors
In mathematics, Clifford's theorem on special divisors is a result of William K. Clifford on algebraic curves, showing the constraints on special linear systems on a curve C.
Chow's moving lemma
In algebraic geometry, Chow's moving lemma, proved by Wei-Liang Chow, states: given algebraic cycles Y, Z on a nonsingular quasi-projective variety X, there is another algebraic cycle Z' on X such tha
Ramanujam vanishing theorem
In algebraic geometry, the Ramanujam vanishing theorem is an extension of the Kodaira vanishing theorem due to Ramanujam, that in particular gives conditions for the vanishing of first cohomology grou