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71-560: [REDACTED] Look up regular  or regularity in Wiktionary, the free dictionary. Regular may refer to: Arts, entertainment, and media [ edit ] Music [ edit ] "Regular" (Badfinger song) Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses in arts, entertainment, and media [ edit ] Regular character ,

142-420: A # = ϕ {\displaystyle {\phi ^{a}}^{\#}=\phi } as well as f # a = f . {\displaystyle {f^{\#}}^{a}=f.} In particular, f is an isomorphism of affine varieties if and only if f is an isomorphism of the coordinate rings. For example, if X is a closed subvariety of an affine variety Y and f

213-478: A 0 : ⋯ : a m ) = ( 1 : a 1 / a 0 : ⋯ : a m / a 0 ) ∼ ( a 1 / a 0 , … , a m / a 0 ) {\displaystyle (a_{0}:\dots :a_{m})=(1:a_{1}/a_{0}:\dots :a_{m}/a_{0})\sim (a_{1}/a_{0},\dots ,a_{m}/a_{0})} . Thus, by definition,

284-472: A concept that has developed to overcome certain difficulties in formally defining conditional probabilities for continuous probability distributions Regular stochastic matrix , a stochastic matrix such that all the entries of some power of the matrix are positive Topology [ edit ] Free regular set , a subset of a topological space that is acted upon disjointly under a given group action Regular homotopy Regular isotopy in knot theory,

355-472: A concept that has developed to overcome certain difficulties in formally defining conditional probabilities for continuous probability distributions Regular stochastic matrix , a stochastic matrix such that all the entries of some power of the matrix are positive Topology [ edit ] Free regular set , a subset of a topological space that is acted upon disjointly under a given group action Regular homotopy Regular isotopy in knot theory,

426-453: A continuous function between metric spaces which preserves Cauchy sequences Regular functions , functions that are analytic and single-valued (unique) in a given region Regular measure , a measure for which every measurable set is "approximately open" and "approximately closed" The regular part , of a Laurent series, the series of terms with positive powers Regular singular points , in theory of ordinary differential equations where

497-453: A continuous function between metric spaces which preserves Cauchy sequences Regular functions , functions that are analytic and single-valued (unique) in a given region Regular measure , a measure for which every measurable set is "approximately open" and "approximately closed" The regular part , of a Laurent series, the series of terms with positive powers Regular singular points , in theory of ordinary differential equations where

568-562: A generalization of a regular polygon to higher dimensions Regular skew polyhedron Logic, set theory, and foundations [ edit ] Axiom of Regularity , also called the Axiom of Foundation, an axiom of set theory asserting the non-existence of certain infinite chains of sets Partition regularity Regular cardinal , a cardinal number that is equal to its cofinality Regular modal logic Probability and statistics [ edit ] Regular conditional probability ,

639-510: A generalization of a regular polygon to higher dimensions Regular skew polyhedron Logic, set theory, and foundations [ edit ] Axiom of Regularity , also called the Axiom of Foundation, an axiom of set theory asserting the non-existence of certain infinite chains of sets Partition regularity Regular cardinal , a cardinal number that is equal to its cofinality Regular modal logic Probability and statistics [ edit ] Regular conditional probability ,

710-455: A main character who appears more frequently and/or prominently than a recurring character Regular division of the plane , a series of drawings by the Dutch artist M. C. Escher which began in 1936 Language [ edit ] Regular inflection , the formation of derived forms such as plurals in ways that are typical for the language Regular verb Regular script , the newest of

781-403: A main character who appears more frequently and/or prominently than a recurring character Regular division of the plane , a series of drawings by the Dutch artist M. C. Escher which began in 1936 Language [ edit ] Regular inflection , the formation of derived forms such as plurals in ways that are typical for the language Regular verb Regular script , the newest of

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852-424: A map f : X → Y between two varieties is regular at a point x if there is a neighbourhood U of x and a neighbourhood V of f ( x ) such that f ( U ) ⊂ V and the restricted function f : U → V is regular as a function on some affine charts of U and V . Then f is called regular , if it is regular at all points of X . The composition of regular maps is again regular; thus, algebraic varieties form

923-458: A morphism f : X → P 1 {\displaystyle f:X\to \mathbf {P} ^{1}} . The important fact is: Theorem  —  Let f : X → Y be a dominating (i.e., having dense image) morphism of algebraic varieties, and let r = dim X  − dim Y . Then Corollary  —  Let f : X → Y be a morphism of algebraic varieties. For each x in X , define Then e

994-414: A morphism from a projective variety to a projective space. Let x be a point of X . Then some i -th homogeneous coordinate of f ( x ) is nonzero; say, i = 0 for simplicity. Then, by continuity, there is an open affine neighborhood U of x such that is a morphism, where y i are the homogeneous coordinates. Note the target space is the affine space A through the identification (

1065-478: A person who visits the same restaurant, pub, store, or transit provider frequently Regular (footedness) in boardsports, a stance in which the left foot leads See also [ edit ] All pages with titles beginning with regular All pages with titles containing regular Irregular (disambiguation) Regular set (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

1136-478: A person who visits the same restaurant, pub, store, or transit provider frequently Regular (footedness) in boardsports, a stance in which the left foot leads See also [ edit ] All pages with titles beginning with regular All pages with titles containing regular Irregular (disambiguation) Regular set (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

1207-420: A power of 60 Regular p-group , a concept capturing some of the more important properties of abelian p -groups, but general enough to include most "small" p -groups Regular prime , a prime number p > 2 that does not divide the class number of the p -th cyclotomic field The regular representation of a group G, the linear representation afforded by the group action of G on itself Regular ring ,

1278-420: A power of 60 Regular p-group , a concept capturing some of the more important properties of abelian p -groups, but general enough to include most "small" p -groups Regular prime , a prime number p > 2 that does not divide the class number of the p -th cyclotomic field The regular representation of a group G, the linear representation afforded by the group action of G on itself Regular ring ,

1349-405: A projective variety X ¯ {\displaystyle {\overline {X}}} ; the difference being that f i 's are in the homogeneous coordinate ring of X ¯ {\displaystyle {\overline {X}}} . Note : The above does not say a morphism from a projective variety to a projective space is given by a single set of polynomials (unlike

1420-527: A religious order subject to a rule of life Regular Force for usage in the Canadian Forces Regular Masonic jurisdictions , or regularity , refers to the constitutional mechanism by which Freemasonry Grand Lodges or Grand Orients give one another mutual recognition People [ edit ] Moses Regular (born 1971), America football player Science and social science [ edit ] Regular bowel movements,

1491-440: A religious order subject to a rule of life Regular Force for usage in the Canadian Forces Regular Masonic jurisdictions , or regularity , refers to the constitutional mechanism by which Freemasonry Grand Lodges or Grand Orients give one another mutual recognition People [ edit ] Moses Regular (born 1971), America football player Science and social science [ edit ] Regular bowel movements,

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1562-471: A ring such that all its localizations have Krull dimension equal to the minimal number of generators of the maximal ideal von Neumann regular ring , or absolutely flat ring (unrelated to the previous sense) Regular semi-algebraic systems in computer algebra Regular semigroup , related to the previous sense *-regular semigroup Analysis [ edit ] Borel regular measure Cauchy-regular function (or Cauchy-continuous function ,)

1633-471: A ring such that all its localizations have Krull dimension equal to the minimal number of generators of the maximal ideal von Neumann regular ring , or absolutely flat ring (unrelated to the previous sense) Regular semi-algebraic systems in computer algebra Regular semigroup , related to the previous sense *-regular semigroup Analysis [ edit ] Borel regular measure Cauchy-regular function (or Cauchy-continuous function ,)

1704-442: A symmetric tessellation of a closed surface Regular matroid , a matroid which can be represented over any field Regular paperfolding sequence , also known as the dragon curve sequence Regular tree grammar Geometry [ edit ] Castelnuovo–Mumford regularity of a coherent sheaf Closed regular sets in solid modeling Irregularity of a surface in algebraic geometry Regular curves Regular grid ,

1775-442: A symmetric tessellation of a closed surface Regular matroid , a matroid which can be represented over any field Regular paperfolding sequence , also known as the dragon curve sequence Regular tree grammar Geometry [ edit ] Castelnuovo–Mumford regularity of a coherent sheaf Closed regular sets in solid modeling Irregularity of a surface in algebraic geometry Regular curves Regular grid ,

1846-456: A tesselation of Euclidean space by congruent bricks Regular map (algebraic geometry) , a map between varieties given by polynomials Regular point, a non-singular point of an algebraic variety Regular point of a differentiable map, a point at which a map is a submersion Regular polygons , polygons with all sides and angles equal Regular polyhedron , a generalization of a regular polygon to higher dimensions Regular polytope ,

1917-456: A tesselation of Euclidean space by congruent bricks Regular map (algebraic geometry) , a map between varieties given by polynomials Regular point, a non-singular point of an algebraic variety Regular point of a differentiable map, a point at which a map is a submersion Regular polygons , polygons with all sides and angles equal Regular polyhedron , a generalization of a regular polygon to higher dimensions Regular polytope ,

1988-449: A uniform distribution of distances between codewords Regular expression , a type of pattern describing a set of strings in computer science Regular graph , a graph such that all the degrees of the vertices are equal Szemerédi regularity lemma , some random behaviors in large graphs Regular language , a formal language recognizable by a finite state automaton (related to the regular expression) Regular map (graph theory) ,

2059-449: A uniform distribution of distances between codewords Regular expression , a type of pattern describing a set of strings in computer science Regular graph , a graph such that all the degrees of the vertices are equal Szemerédi regularity lemma , some random behaviors in large graphs Regular language , a formal language recognizable by a finite state automaton (related to the regular expression) Regular map (graph theory) ,

2130-396: Is continuous with respect to Zariski topologies on the source and the target. The image of a morphism of varieties need not be open nor closed (for example, the image of A 2 → A 2 , ( x , y ) ↦ ( x , x y ) {\displaystyle \mathbf {A} ^{2}\to \mathbf {A} ^{2},\,(x,y)\mapsto (x,xy)}

2201-405: Is upper-semicontinuous ; i.e., for each integer n , the set is closed. In Mumford's red book, the theorem is proved by means of Noether's normalization lemma . For an algebraic approach where the generic freeness plays a main role and the notion of " universally catenary ring " is a key in the proof, see Eisenbud, Ch. 14 of "Commutative algebra with a view toward algebraic geometry." In fact,

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2272-465: Is a normal variety , then f is biregular. (cf. Zariski's main theorem .) A regular map between complex algebraic varieties is a holomorphic map . (There is actually a slight technical difference: a regular map is a meromorphic map whose singular points are removable , but the distinction is usually ignored in practice.) In particular, a regular map into the complex numbers is just a usual holomorphic function (complex-analytic function). Let be

2343-576: Is a finite surjective morphism, if X , Y are complete and F a coherent sheaf on Y , then from the Leray spectral sequence H p ⁡ ( Y , R q f ∗ f ∗ F ) ⇒ H p + q ⁡ ( X , f ∗ F ) {\displaystyle \operatorname {H} ^{p}(Y,R^{q}f_{*}f^{*}F)\Rightarrow \operatorname {H} ^{p+q}(X,f^{*}F)} , one gets: In particular, if F

2414-591: Is a morphism, then writing ϕ = f # {\displaystyle \phi =f^{\#}} , we need to show where m x , m f ( x ) {\displaystyle {\mathfrak {m}}_{x},{\mathfrak {m}}_{f(x)}} are the maximal ideals corresponding to the points x and f ( x ); i.e., m x = { g ∈ k [ X ] ∣ g ( x ) = 0 } {\displaystyle {\mathfrak {m}}_{x}=\{g\in k[X]\mid g(x)=0\}} . This

2485-424: Is a regular map X → P . In particular, when X is a smooth complete curve, any rational function on X may be viewed as a morphism X → P and, conversely, such a morphism as a rational function on X . On a normal variety (in particular, a smooth variety ), a rational function is regular if and only if it has no poles of codimension one. This is an algebraic analog of Hartogs' extension theorem . There

2556-446: Is a tensor power L ⊗ n {\displaystyle L^{\otimes n}} of a line bundle, then R q f ∗ ( f ∗ F ) = R q f ∗ O X ⊗ L ⊗ n {\displaystyle R^{q}f_{*}(f^{*}F)=R^{q}f_{*}{\mathcal {O}}_{X}\otimes L^{\otimes n}} and since

2627-409: Is also a relative version of this fact; see [2] . A morphism between algebraic varieties that is a homeomorphism between the underlying topological spaces need not be an isomorphism (a counterexample is given by a Frobenius morphism t ↦ t p {\displaystyle t\mapsto t^{p}} .) On the other hand, if f is bijective birational and the target space of f

2698-485: Is an algebra homomorphism, then it induces the morphism given by: writing k [ Y ] = k [ y 1 , … , y m ] / J , {\displaystyle k[Y]=k[y_{1},\dots ,y_{m}]/J,} where y ¯ i {\displaystyle {\overline {y}}_{i}} are the images of y i {\displaystyle y_{i}} 's. Note ϕ

2769-476: Is immediate.) This fact means that the category of affine varieties can be identified with a full subcategory of affine schemes over k . Since morphisms of varieties are obtained by gluing morphisms of affine varieties in the same way morphisms of schemes are obtained by gluing morphisms of affine schemes, it follows that the category of varieties is a full subcategory of the category of schemes over k . For more details, see [1] . A morphism between varieties

2840-421: Is induced by a dominant rational map from X to Y . Hence, the above construction determines a contravariant-equivalence between the category of algebraic varieties over a field k and dominant rational maps between them and the category of finitely generated field extension of k . If X is a smooth complete curve (for example, P ) and if f is a rational map from X to a projective space P , then f

2911-433: Is injective. Thus, the dominant map f induces an injection on the level of function fields: where the direct limit runs over all nonempty open affine subsets of Y . (More abstractly, this is the induced map from the residue field of the generic point of Y to that of X .) Conversely, every inclusion of fields k ( Y ) ↪ k ( X ) {\displaystyle k(Y)\hookrightarrow k(X)}

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2982-578: Is neither open nor closed). However, one can still say: if f is a morphism between varieties, then the image of f contains an open dense subset of its closure (cf. constructible set ). A morphism f : X → Y of algebraic varieties is said to be dominant if it has dense image. For such an f , if V is a nonempty open affine subset of Y , then there is a nonempty open affine subset U of X such that f ( U ) ⊂ V and then f # : k [ V ] → k [ U ] {\displaystyle f^{\#}:k[V]\to k[U]}

3053-426: Is the inclusion, then f is the restriction of regular functions on Y to X . See #Examples below for more examples. In the particular case that Y equals A the regular maps f : X → A are called regular functions , and are algebraic analogs of smooth functions studied in differential geometry. The ring of regular functions (that is the coordinate ring or more abstractly the ring of global sections of

3124-399: Is the restriction of a polynomial map A n → A m {\displaystyle \mathbb {A} ^{n}\to \mathbb {A} ^{m}} . Explicitly, it has the form: where the f i {\displaystyle f_{i}} s are in the coordinate ring of X : where I is the ideal defining X (note: two polynomials f and g define

3195-414: The category of algebraic varieties where the morphisms are the regular maps. Regular maps between affine varieties correspond contravariantly in one-to-one to algebra homomorphisms between the coordinate rings: if f : X → Y is a morphism of affine varieties, then it defines the algebra homomorphism where k [ X ] , k [ Y ] {\displaystyle k[X],k[Y]} are

3266-454: The pre-images of prime ideals . All morphisms between affine schemes are of this type and gluing such morphisms gives a morphism of schemes in general. Now, if X , Y are affine varieties; i.e., A , B are integral domains that are finitely generated algebras over an algebraically closed field k , then, working with only the closed points, the above coincides with the definition given at #Definition . (Proof: If f  : X → Y

3337-754: The Chinese script styles Mathematics [ edit ] Algebra and number theory [ edit ] Regular category , a kind of category that has similarities to both Abelian categories and to the category of sets Regular chains in computer algebra Regular element (disambiguation) , certain kinds of elements of an algebraic structure Regular extension of fields Regular ideal (multiple definitions) Regular Lie group Regular matrix (disambiguation) Regular monomorphisms and regular epimorphisms , monomorphisms (resp. epimorphisms) which equalize (resp. coequalize) some parallel pair of morphisms Regular numbers , numbers which evenly divide

3408-699: The Chinese script styles Mathematics [ edit ] Algebra and number theory [ edit ] Regular category , a kind of category that has similarities to both Abelian categories and to the category of sets Regular chains in computer algebra Regular element (disambiguation) , certain kinds of elements of an algebraic structure Regular extension of fields Regular ideal (multiple definitions) Regular Lie group Regular matrix (disambiguation) Regular monomorphisms and regular epimorphisms , monomorphisms (resp. epimorphisms) which equalize (resp. coequalize) some parallel pair of morphisms Regular numbers , numbers which evenly divide

3479-434: The affine case). For example, let X be the conic y 2 = x z {\displaystyle y^{2}=xz} in P . Then two maps ( x : y : z ) ↦ ( x : y ) {\displaystyle (x:y:z)\mapsto (x:y)} and ( x : y : z ) ↦ ( y : z ) {\displaystyle (x:y:z)\mapsto (y:z)} agree on

3550-436: The condition is for some pair ( g , h ) not for all pairs ( g , h ); see Examples . If X is a quasi-projective variety ; i.e., an open subvariety of a projective variety, then the function field k ( X ) is the same as that of the closure X ¯ {\displaystyle {\overline {X}}} of X and thus a rational function on X is of the form g / h for some homogeneous elements g , h of

3621-465: The coordinate rings of X and Y ; it is well-defined since g ∘ f = g ( f 1 , … , f m ) {\displaystyle g\circ f=g(f_{1},\dots ,f_{m})} is a polynomial in elements of k [ X ] {\displaystyle k[X]} . Conversely, if ϕ : k [ Y ] → k [ X ] {\displaystyle \phi :k[Y]\to k[X]}

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3692-473: The equivalence relation of link diagrams that is generated by using the 2nd and 3rd Reidemeister moves only Regular space (or T 3 {\displaystyle T_{3}} ) space, a topological space in which a point and a closed set can be separated by neighborhoods Organizations [ edit ] Regular army for military usage Regular Baptists , an 18th-century American and Canadian Baptist group Regular clergy , members of

3763-473: The equivalence relation of link diagrams that is generated by using the 2nd and 3rd Reidemeister moves only Regular space (or T 3 {\displaystyle T_{3}} ) space, a topological space in which a point and a closed set can be separated by neighborhoods Organizations [ edit ] Regular army for military usage Regular Baptists , an 18th-century American and Canadian Baptist group Regular clergy , members of

3834-485: The 💕 [REDACTED] Look up regular  or regularity in Wiktionary, the free dictionary. Regular may refer to: Arts, entertainment, and media [ edit ] Music [ edit ] "Regular" (Badfinger song) Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses in arts, entertainment, and media [ edit ] Regular character ,

3905-558: The growth of solutions is bounded by an algebraic function Regularity, the degree of differentiability of a function Regularity conditions arise in the study of first-class constraints in Hamiltonian mechanics Regularity of an elliptic operator Regularity theory of elliptic partial differential equations Combinatorics, discrete math, and mathematical computer science [ edit ] Regular algebra , or Kleene algebra Regular code , an algebraic code with

3976-498: The growth of solutions is bounded by an algebraic function Regularity, the degree of differentiability of a function Regularity conditions arise in the study of first-class constraints in Hamiltonian mechanics Regularity of an elliptic operator Regularity theory of elliptic partial differential equations Combinatorics, discrete math, and mathematical computer science [ edit ] Regular algebra , or Kleene algebra Regular code , an algebraic code with

4047-462: The homogeneous coordinates, for all x in U and by continuity for all x in X as long as the f i 's do not vanish at x simultaneously. If they vanish simultaneously at a point x of X , then, by the above procedure, one can pick a different set of f i 's that do not vanish at x simultaneously (see Note at the end of the section.) In fact, the above description is valid for any quasi-projective variety X , an open subvariety of

4118-513: The open subset { ( x : y : z ) ∈ X ∣ x ≠ 0 , z ≠ 0 } {\displaystyle \{(x:y:z)\in X\mid x\neq 0,z\neq 0\}} of X (since ( x : y ) = ( x y : y 2 ) = ( x y : x z ) = ( y : z ) {\displaystyle (x:y)=(xy:y^{2})=(xy:xz)=(y:z)} ) and so defines

4189-442: The opposite of constipation Regular economy , an economy characterized by an excess demand function whose slope at any equilibrium price vector is non-zero Regular moon , a natural satellite that has low eccentricity and a relatively close and prograde orbit Regular solutions in chemistry, solutions that diverge from the behavior of an ideal solution only moderately Other uses [ edit ] Regular customer ,

4260-442: The opposite of constipation Regular economy , an economy characterized by an excess demand function whose slope at any equilibrium price vector is non-zero Regular moon , a natural satellite that has low eccentricity and a relatively close and prograde orbit Regular solutions in chemistry, solutions that diverge from the behavior of an ideal solution only moderately Other uses [ edit ] Regular customer ,

4331-447: The proof there shows that if f is flat , then the dimension equality in 2. of the theorem holds in general (not just generically). Let f : X → Y be a finite surjective morphism between algebraic varieties over a field k . Then, by definition, the degree of f is the degree of the finite field extension of the function field k ( X ) over f k ( Y ). By generic freeness , there is some nonempty open subset U in Y such that

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4402-467: The restriction f | U is given by where g i 's are regular functions on U . Since X is projective, each g i is a fraction of homogeneous elements of the same degree in the homogeneous coordinate ring k [ X ] of X . We can arrange the fractions so that they all have the same homogeneous denominator say f 0 . Then we can write g i = f i / f 0 for some homogeneous elements f i 's in k [ X ]. Hence, going back to

4473-478: The restriction of the structure sheaf O X to f ( U ) is free as O Y | U -module . The degree of f is then also the rank of this free module. If f is étale and if X , Y are complete , then for any coherent sheaf F on Y , writing χ for the Euler characteristic , (The Riemann–Hurwitz formula for a ramified covering shows the "étale" here cannot be omitted.) In general, if f

4544-400: The same degree in k [ X ¯ ] {\displaystyle k[{\overline {X}}]} such that f = g / h and h does not vanish at x . This characterization is sometimes taken as the definition of a regular function. If X = Spec A and Y = Spec B are affine schemes , then each ring homomorphism ϕ : B → A determines a morphism by taking

4615-404: The same degree in the homogeneous coordinate ring k [ X ¯ ] {\displaystyle k[{\overline {X}}]} of X ¯ {\displaystyle {\overline {X}}} (cf. Projective variety#Variety structure .) Then a rational function f on X is regular at a point x if and only if there are some homogeneous elements g , h of

4686-425: The same function on X if and only if f  −  g is in I ). The image f ( X ) lies in Y , and hence satisfies the defining equations of Y . That is, a regular map f : X → Y {\displaystyle f:X\to Y} is the same as the restriction of a polynomial map whose components satisfy the defining equations of Y {\displaystyle Y} . More generally,

4757-490: The structure of a locally ringed space ; a morphism between algebraic varieties is precisely a morphism of the underlying locally ringed spaces. If X and Y are closed subvarieties of A n {\displaystyle \mathbb {A} ^{n}} and A m {\displaystyle \mathbb {A} ^{m}} (so they are affine varieties ), then a regular map f : X → Y {\displaystyle f\colon X\to Y}

4828-506: The structure sheaf) is a fundamental object in affine algebraic geometry. The only regular function on a projective variety is constant (this can be viewed as an algebraic analogue of Liouville's theorem in complex analysis ). A scalar function f : X → A is regular at a point x if, in some open affine neighborhood of x , it is a rational function that is regular at x ; i.e., there are regular functions g , h near x such that f = g / h and h does not vanish at x . Caution:

4899-412: The support of R q f ∗ O X {\displaystyle R^{q}f_{*}{\mathcal {O}}_{X}} has positive codimension if q is positive, comparing the leading terms, one has: (since the generic rank of f ∗ O X {\displaystyle f_{*}{\mathcal {O}}_{X}} is the degree of f .) If f

4970-455: The title Regular . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Regular&oldid=1244587866 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages regular From Misplaced Pages,

5041-482: The title Regular . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Regular&oldid=1244587866 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Regular function An algebraic variety has naturally

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