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I'm organizing theorems in Algebraic Geometry that are listed in Wikipedia. Wikipedia: Theorems in Algebraic Geometry Abhyankar's lemma allows one to kill tame ramification by taking an extension of a base field. More precisely, Abhyankar's lemma states that if A, B, C are local fields such that A and B are finite extensions of C, with ramification indices a and b, and B is tamely ramified over C and b divides a, then the compositum AB is an unramified extension of A. To 'classify' addition theorems it is necessary to put some restriction on the type of function G admitted, such that F(x + y) = G(F(x), F(y)). In this identity one can assume that F and G are vectorvalued (have several components). An algebraic addition theorem is one in which G can be taken to be a vector of polynomials?, in some set of variables. The conclusion of the mathematicians of the time was that the theory of abelian functions? essentially exhausted the interesting possibilities: considered as a functional equation? to be solved with polynomials, or indeed rational functions? or algebraic functions?, there were no further types of solution. Behrend's formula is a generalization of the Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite field. Algebraic curves Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves, which do not share a common component (that is, which do not have infinitely many common points). The theorem claims that the number of common points of two such curves is at most equal to the product of their degrees, and equality holds if one counts points at infinity, points with complex coordinates (or more generally, coordinates from the algebraic closure of the ground field), and if each point is counted with its intersection multiplicity. Belyi's theorem on algebraic curves states that any nonsingular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only. 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 algebraic curve. AF+BG theorem (Max Noether's fundamental theorem) describes when the equation of an algebraic curve in the complex projective plane can be written in terms of the equations of two other algebraic curves. Abhyankar–Moh theorem states that if {$ \displaystyle L $} is a complex line in the complex affine plane {$ \displaystyle \mathbb {C} ^{2} $}, then every embedding of {$ \displaystyle L $} into {$ \displaystyle \mathbb {C} ^{2} $} extends to an automorphism of the plane. More generally, the same theorem applies to lines and planes over any algebraically closed field of characteristic zero, and to certain wellbehaved subsets of higherdimensional complex affine spaces. Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane P2. Every cubic curve C1 on an algebraically closed field that passes through a given set of eight points P1, ..., P8 also passes through a certain (fixed) ninth point P9, counting multiplicities. Chasles–Cayley–Brill formula (also known as the CayleyBrill formula) states that a correspondence T of valence k from an algebraic curve C of genus g to itself has d + e + 2kg united points, where d and e are the degrees of T and its inverse. 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 calculated from the degrees of the initial two pencils. 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. The simplest is that the automorphism group of X is finite (see though Hurwitz's automorphisms theorem). More generally,
Enriques–Babbage theorem states that a canonical curve is either a settheoretic intersection of quadrics, or trigonal, or a plane quintic. Faltings's theorem states that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. It was later generalized by replacing Q by any number field. Gudkov's conjecture is now a theorem, which states that "a Mcurve* of even degree 2d obeys p – n ≡ d2 (mod 8)", where p is the number of positive ovals and n the number of negative ovals of the Mcurve. Harnack's curve theorem describes the possible numbers of connected components that an algebraic curve can have, in terms of the degree of the curve. 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 spanned by such curves (up to linear equivalence) has a onedimensional subspace on which it is positive definite (not uniquely determined), and decomposes as a direct sum of some such onedimensional subspace, and a complementary subspace on which it is negative definite. Reiss relation is a condition on the secondorder elements of the points of a plane algebraic curve meeting a given line. Torelli theorem is a classical result of algebraic geometry over the complex number field, stating that a nonsingular projective algebraic curve (compact Riemann surface) C is determined by its Jacobian variety J(C), when the latter is given in the form of a principally polarized abelian variety. In other words, the complex torus J(C), with certain 'markings', is enough to recover C. The same statement holds over any algebraically closed field. Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasialgebraically closed (i.e., C1). This implies that the Brauer group of any such field vanishes,[1] and more generally that all the Galois cohomology groups H i(K, K*) vanish for i ≥ 1. This result is used to calculate the étale cohomology groups of an algebraic curve. Weber's theorem. Consider two nonsingular curves C and C′ having the same genus g > 1. If there is a rational correspondence φ between C and C′, then φ is a birational transformation. Weil reciprocity law is a result 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. rational functions on C, then f((g)) = g((f)) where the notation has this meaning: (h) is the divisor of the function h, or in other words the formal sum of its zeroes and poles counted with multiplicity; and a function applied to a formal sum means the product (with multiplicities, poles counting as a negative multiplicity) of the values of the function at the points of the divisor. With this definition there must be the sidecondition, that the divisors of f and g have disjoint support (which can be removed). Elliptic curves Modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms. The theorem states that any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve {$ X_{0}(N) $} for some integer N. Néron–Ogg–Shafarevich criterion states that if A is an elliptic curve or abelian variety over a local field K and ℓ is a prime not dividing the characteristic of the residue field of K then A has good reduction if and only if the ℓadic Tate module Tℓ of A is unramified. Raynaud's isogeny theorem relates the Faltings heights of two isogeneous elliptic curves. Vector bundles, line bundles Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over {$ \displaystyle \mathbb {CP} ^{1} $} is a direct sum of holomorphic line bundles. Appell–Humbert theorem describes the line bundles on a complex torus or complex abelian variety. Lange's conjecture is a conjecture about stability of vector bundles over curves. Lefschetz theorem on (1,1)classes, named after Solomon Lefschetz, is a classical statement relating holomorphic line bundles on a compact Kähler manifold to classes in its integral cohomology. Reider's theorem gives conditions for a line bundle on a projective surface to be very ample. Group action on variety Borel fixedpoint theorem is a fixedpoint theorem in algebraic geometry generalizing the Lie–Kolchin theorem. If G is a connected, solvable, algebraic group acting regularly on a nonempty, complete algebraic variety V over an algebraically closed field k, then there is a G fixedpoint of V. Luna's slice theorem describes the local behavior of an action of a reductive algebraic group on an affine variety. Sumihiro's theorem states that a normal algebraic variety with an action of a torus can be covered by torusinvariant affine open subsets. Cohomology Borel's theorem says the cohomology ring of a classifying space or a classifying stack is a polynomial ring. Atiyah–Bott formula says the cohomology ring {$ \operatorname {H}^{*}(\operatorname {Bun}_{G}(X),{\mathbb {Q}}_{l}) $} of the moduli stack of principal bundles is a free gradedcommutative algebra on certain homogeneous generators. 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 manifold. 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. There are several generalizations: the Frobenius endomorphism can be replaced by a more general endomorphism, in which case the points over a finite field are replaced by its fixed points, and there is also a more general version for a sheaf over the variety, where the cohomology groups are replaced by cohomology with coefficients in the sheaf. Grothendieck–Riemann–Roch theorem is a farreaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces. Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result contributing to the Riemann–Roch problem for complex algebraic varieties of all dimensions. It was the first successful generalisation of the classical Riemann–Roch theorem on Riemann surfaces to all higher dimensions, and paved the way to the Grothendieck–Hirzebruch–Riemann–Roch theorem proved about three years later. Holomorphic Lefschetz formula is an analogue for complex manifolds of the Lefschetz fixedpoint formula that relates a sum over the fixed points of a holomorphic vector field of a compact complex manifold to a sum over its Dolbeault cohomology groups. Kawamata–Viehweg vanishing theorem is an extension of the Kodaira vanishing theorem, on the vanishing of coherent cohomology groups, to logarithmic pairs. The theorem states that if L is a big nef line bundle (for example, an ample line bundle) on a complex projective manifold with canonical line bundle K, then the coherent cohomology groups Hi(L⊗K) vanish for all positive i. Ramanujam vanishing theorem is an extension of the Kodaira vanishing theorem that in particular gives conditions for the vanishing of first cohomology groups of coherent sheaves on a surface. Kempf vanishing theorem 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 reductive algebraic group over an algebraically closed field, B a Borel subgroup, and L(λ) a line bundle associated to λ. In characteristic 0 this is a special case of the Borel–Weil–Bott theorem, but unlike the Borel–Weil–Bott theorem, the Kempf vanishing theorem still holds in positive characteristic. Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are automatically zero. The implications for the group with index q = 0 is usually that its dimension — the number of independent global sections — coincides with a holomorphic Euler characteristic that can be computed using the HirzebruchRiemannRoch theorem. Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those of Y. Leray's theorem relates abstract sheaf cohomology with Čech cohomology. Let F be a sheaf on a topological space X and U an open cover of X . If F is acyclic on every finite intersection of elements of U, then {$ {\check {H}}^{q}({\mathcal {U}},{\mathcal {F}})=H^{q}(X,{\mathcal {F}}),$} {$ {\check {H}}^{q}({\mathcal {U}},{\mathcal {F}})=H^{q}(X,{\mathcal {F}}),$} where {$ \displaystyle {\check {H}}^{q}({\mathcal {U}},{\mathcal {F}}) $} {$ \displaystyle {\check {H}}^{q}({\mathcal {U}},{\mathcal {F}}) $} is the qth Čech cohomology group of F with respect to the open cover U. Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an ndimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the (n − k)th homology group of M, for all integers k {$ H^{k}(M)\cong H_{nk}(M). $} Proper base change theorem states the following: let {$ f:X\to S $} be a proper morphism between noetherian schemes, and F Sflat coherent sheaf on X. If {$ S=\operatorname {Spec} A $}, then there is a finite complex {$ 0\to K^{0}\to K^{1}\to \cdots \to K^{n}\to 0 $} of finitely generated projective Amodules and a natural isomorphism of functors {$ H^{p}(X\times _{S}\operatorname {Spec} ,{\mathcal {F}}\otimes _{A})\to H^{p}(K^{\bullet }\otimes _{A}),p\geq 0 $} on the category of Aalgebras. The proper base change theorem of étale cohomology states that the higher direct image {$ R^{i}f_{*}{\mathcal {F}} $} of a torsion sheaf F along a proper morphism f commutes with base change. A closely related, the finiteness theorem states that the étale cohomology groups of a constructible sheaf on a complete variety are finite. Theorem (finiteness): Let X be a variety over a separably closed field and F a constructible sheaf on {$ X_{\text{et}}$}. Then {$ H^{r}(X,{\mathcal {F}}) $} are finite in each of the following cases: (i) X is complete, or (ii) F has no ptorsion, where p is the characteristic of k. Sheaves Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X.
Mumford vanishing theorem states that if L is a semiample invertible sheaf with Iitaka dimension at least 2 on a complex projective manifold, then {$ H^{i}(X,L^{1})=0{\text{ for }}i=0,1.\ $} Projection formula states that,[1][2] for a quasicompact separated morphism of schemes {$ f:X\to Y $}, a quasicoherent sheaf F on X, a locally free sheaf E on Y, the natural maps of sheaves {$ R^{i}f_{*}{\mathcal {F}}\otimes {\mathcal {E}}\to R^{i}f_{*}({\mathcal {F}}\otimes f^{*}{\mathcal {E}}) $} are isomorphisms. Spectrum Chevalley–Iwahori–Nagata theorem states that if a linear algebraic group G is acting linearly on a finitedimensional vector space V, then the map from V/G to the spectrum of the ring of invariant polynomials is an isomorphism if this ring is finitely generated and all orbits of G on V are closed Grothendieck's connectedness theorem states that if A is a complete local ring whose spectrum is kconnected and f is in the maximal ideal, then Spec(A/fA) is (k − 1)connected. Algebraic groups Chevalley's structure theorem states that a smooth connected algebraic group over a perfect field has a unique normal smooth connected affine algebraic subgroup such that the quotient is an abelian variety. Schemes Chow's lemma A proper morphism is fairly close to being a projective morphism. If X is a scheme that is proper over a noetherian base S, then there exists a projective scheme {$ X' $} and a surjective {$ S $} morphism {$ f\colon X'\to X $} that induces an isomorphism {$ f^{1}(U)\simeq U $} for some dense open {$ U\subseteq X $}. Grothendieck existence theorem gives conditions that enable one to lift infinitesimal deformations of a scheme to a deformation, and to lift schemes over infinitesimal neighborhoods over a subscheme of a scheme S to schemes over S. Regular Embedding A closed immersion {$ i:X\hookrightarrow Y $} of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of {$ X\cap U $} is generated by a regular sequence of length r. Serre–Tate theorem says that under certain conditions an abelian scheme and its pdivisible group have the same infinitesimal deformation theory. Theorem on formal functions states the following: Let {$ f:X\to S $} be a proper morphism of noetherian schemes with a coherent sheaf F on X. Let {$ S_{0} $} be a closed subscheme of S defined by I and {$ {\widehat {X}},{\widehat {S}} $} formal completions with respect to {$ X_{0}=f^{1}(S_{0}) $} and {$ S_{0} $}. Then for each {$ p\geq 0 $} the canonical (continuous) map {$ (R^{p}f_{*}{\mathcal {F}})^{\wedge }\to \varprojlim _{k}R^{p}f_{*}{\mathcal {F}}_{k} $} is an isomorphism of (topological) {$ {\mathcal {O}}_{\widehat {S}} $}modules, where
Algebraic cycles Chow's moving lemma states: given algebraic cycles Y, Z on a nonsingular quasiprojective variety X, there is another algebraic cycle Z' on X such that Z' is rationally equivalent to Z and Y and Z' intersect properly. The lemma is one of key ingredients in developing the intersection theory, as it is used to show the uniqueness of the theory. Clifford's theorem on special divisors is a result of W. K. Clifford (1878) on algebraic curves, showing the constraints on special linear systems on a curve C. For an effective special divisor D, ℓ(D) − 1 ≤ d/2, and the case of equality here is only for D zero or canonical, or C a hyperelliptic curve and D linearly equivalent to an integral multiple of a hyperelliptic divisor. Integral over moduli spaces ELSV formula is an equality between a Hurwitz number (counting ramified coverings of the sphere) and an integral over the moduli space of stable curves. Algebraic sets, varieties, subvarieties Fulton–Hansen connectedness theorem states that if V and W are irreducible algebraic subvarieties of a projective space P, all over an algebraically closed field, and if dim(V) + dim (W) > dim (P) in terms of the dimension of an algebraic variety, then the intersection U of V and W is connected. Gram's theorem states that an algebraic set in a finitedimensional vector space invariant under some linear group can be defined by absolute invariants. 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 correspond to algebraic integers all of whose conjugates (given by eigenvalues of the Frobenius endomorphism on the first cohomology group or Tate module) have absolute value √q. Mnev's universality theorem is a result which can be used to represent algebraic (or semi algebraic) varieties as realizations of oriented matroids. Nagata's compactification theorem implies that every abstract variety can be embedded in a complete variety, and more generally shows that a separated and finite type morphism to a Noetherian scheme S can be factored into an open immersion followed by a proper mapping. Given an algebraic variety (or more generally scheme) X, states that if
then X is affine. Tate's isogeny theorem 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 connectedness theorem says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's main theorem to the case when the morphism of varieties need not be birational. Suppose that f is a proper surjective morphism of varieties from X to Y such that the function field of Y is separably closed in that of X. Then Zariski's connectedness theorem says that the inverse image of any normal point of Y is connected. Polynomial rings Hilbert's Nullstellensatz is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, an important branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. Let k be a field (such as the rational numbers) and K be an algebraically closed field extension (such as the complex numbers), consider the polynomial ring k[X1,X2,..., Xn] and let I be an ideal in this ring. The algebraic set V(I) defined by this ideal consists of all ntuples x = (x1,...,xn) in Kn such that f(x) = 0 for all f in I. Hilbert's Nullstellensatz states that if p is some polynomial in k[X1,X2,..., Xn] that vanishes on the algebraic set V(I), i.e. p(x) = 0 for all x in V(I), then there exists a natural number r such that pr is in I. Kodaira embedding theorem characterises nonsingular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomials. Stengle's Positivstellensatz characterizes polynomials that are positive on a semialgebraic set, which is defined by systems of inequalities of polynomials with real coefficients, or more generally, coefficients from any real closed field. It can be thought of as an ordered analogue of Hilbert's Nullstellensatz. Tarski–Seidenberg theorem states that a set in (n + 1)dimensional space defined by polynomial equations and inequalities can be projected down onto ndimensional space, and the resulting set is still definable in terms of polynomial identities and inequalities. It implies that quantifier elimination is possible over the reals, that is that every formula constructed from polynomial equations and inequalities by logical connectors ∨ (or), ∧ (and), ¬ (not) and quantifiers ∀ (for all), ∃ (exists) is equivalent with a similar formula without quantifiers. An important consequence is the decidability of the theory of realclosed fields. Automorphism groups Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientationpreserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1). A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann surfaces are synonymous with nonsingular complex projective algebraic curves, a Hurwitz surface can also be called a Hurwitz curve. Algebraic spaces Keel–Mori theorem gives conditions for the existence of the quotient of an algebraic space by a group. Vector Kempf–Ness theorem gives a criterion for the stability of a vector in a representation of a complex reductive group. If the complex vector space is given a norm that is invariant under a maximal compact subgroup of the reductive group, then the Kempf–Ness theorem states that a vector is stable if and only if the norm attains a minimum value on the orbit of the vector. Frobenius morphism Lang's theorem, introduced by Serge Lang, states: if G is a connected smooth algebraic group over a finite field {$ {F} _{q}$}, then, writing {$ \displaystyle \sigma :G\to G,\,x\mapsto x^{q} $} for the Frobenius, the morphism of varieties {$ \displaystyle G\to G,\,x\mapsto x^{1}\sigma (x)$} is surjective. Note that the kernel of this map (i.e.,{$ \displaystyle G=G({\overline {\mathbf {F} _{q}}})\to G({\overline {\mathbf {F} _{q}}}) $} is precisely {$ \displaystyle G(\mathbf {F} _{q}) $}. The theorem implies that {$ \displaystyle H^{1}(\mathbf {F} _{q},G)=H_{\mathrm {{\acute {e}}t} }^{1}(\operatorname {Spec} \mathbf {F} _{q},G) $} vanishes, and, consequently, any Gbundle on {$ \displaystyle \operatorname {Spec} \mathbf {F} _{q} $} is isomorphic to the trivial one. Also, the theorem plays a basic role in the theory of finite groups of Lie type. Field extensions Lüroth's theorem asserts that every field that lies between two other fields K and K(X) must be generated as an extension of K by a single element of K(X). Algebraic surfaces Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, giving a criterion for a rational surface. Let S be an algebraic surface that is nonsingular and projective. Suppose there is a morphism φ from S to the projective line, with general fibre also a projective line. Then the theorem states that S is rational. Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. The Riemann–Roch theorem for a compact Riemann surface of genus g with canonical divisor K states {$ \ell (D)\ell (KD)=\deg(D)g+1. $} Kawasaki's Riemann–Roch formula is the Riemann–Roch formula for orbifolds. Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. One form of the Riemann–Roch theorem states that if D is a divisor on a nonsingular projective surface then {$ \chi (D)=\chi (0)+{\tfrac {1}{2}}D.(DK)\, $} where χ is the holomorphic Euler characteristic, the dot . is the intersection number, and K is the canonical divisor. The constant χ(0) is the holomorphic Euler characteristic of the trivial bundle, and is equal to 1 + pa, where pa is the arithmetic genus of the surface. For comparison, the Riemann–Roch theorem for a curve states that χ(D) = χ(0) + deg(D). Degeneracy locus Porteous formula Given a morphism of vector bundles E, F of ranks m and n over a smooth variety, its kth degeneracy locus (k ≤ min(m,n)) is the variety of points where it has rank at most k. If all components of the degeneracy locus have the expected codimension (m – k)(n – k) then Porteous's formula states that its fundamental class is the determinant of the matrix of size m – k whose (i, j) entry is the Chern class cn–k+j–i(F – E). Local ring Ramanujam–Samuel theorem gives conditions for a divisor of a local ring to be principal. Schlessinger's theorem is a theorem in deformation theory that gives conditions for a functor of artinian local rings to be prorepresentable, refining an earlier theorem of Grothendieck. Galois representations Ribet's theorem is a statement in number theory concerning properties of Galois representations associated with modular forms. Let f be a weight 2 newform on Γ0(qN)–i.e. of level qN where q does not divide N–with absolutely irreducible 2dimensional mod p Galois representation ρf,p unramified at q if q ≠ p and finite flat at q = p. Then there exists a weight 2 newform g of level N such that {$ \rho _{f,p}\simeq \rho _{g,p}. $} In particular, if E is an elliptic curve over Q with conductor qN, then the Modularity theorem guarantees that there exists a weight 2 newform f of level qN such that the 2dimensional mod p Galois representation ρf, p of f is isomorphic to the 2dimensional mod p Galois representation ρE, p of E. Hyperplane sections Theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields. Let X be a smooth quasiprojective variety over an algebraically closed field, embedded in a projective space {$ \mathbf {P} ^{n} $}. Let {$ H $} denote the complete system of hyperplane divisors in {$ \mathbf {P} ^{n} $}. Recall that it is the dual space {$ (\mathbf {P} ^{n})^{\star } $} of {$ \mathbf {P} ^{n} $} and is isomorphic to {$ \mathbf {P} ^{n} $}. The theorem of Bertini states that the set of hyperplanes not containing X and with smooth intersection with X contains an open dense subset of the total system of divisors {$ H $}. The set itself is open if X is projective. If dim(X) ≥ 2, then these intersections (called hyperplane sections of X) are connected, hence irreducible. Torsion group The torsion conjecture or uniform boundedness conjecture for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field. Projective spaces Veblen–Young theorem states that a projective space of dimension at least 3 can be constructed as the projective space associated to a vector space over a division ring. NonDesarguesian planes give examples of 2dimensional projective spaces that do not arise from vector spaces over division rings, showing that the restriction to dimension at least 3 is necessary. Birational map Zariski's main theorem is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness theorem when the two varieties are birational. ... The total transform of a normal fundamental point of a birational map has positive dimension. 
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