Unreferenced date December 2009 Dablink For the mathematical meanings, see algebraic variety and universal algebra . In botanical nomenclature, a subvariety subvarietas is a taxonomic rank below that of Variety biology variety varietas but above that of Form botany form forma it is an infraspecific taxon . Its name consists of three parts a genus name, a specific epithet and an infraspecific epithet. To indicate the rank, the abbreviation subvar. should be put before the infraspecific epithet. Category Botanical nomenclature rank27 ... more details
Plant variety may refer to Variety botany , a taxonomic nomenclature rank in botany, below subspecies, but above subvariety and form Plant variety law , a non taxonomic, exclusively legal term applied to plants for which patent protection has been applied or to which it applies taxonomic categorization of such a plant may, on a case by case basis, be any Infraspecies infraspecific rank , usually a cultivar or hybrid Variety , an informal, incorrect and ambiguous substitute for form botany , a taxonomic nomenclature rank in botany, below variety as formally defined at variety botany and subvariety but above subform Variety , an informal, incorrect, ambiguous and vague substitute for cultivar or hybrid biology , the lowest taxonomic nomenclature ranks in botany used especially with regard to grapes and rice the equivalent term varietal , though not an official botany term, is also common in horticulture generally and is not as ambiguous, although still vague disambig id Varietas ... more details
In mathematics , specifically algebraic geometry , an exceptional divisor for a regular map algebraic geometry regular map math f X rightarrow Y math of varieties is a kind of large subvariety of math X math which is crushed by math f math , in a certain definite sense. More strictly, f has an associated exceptional locus which describes how it identifies nearby points in codimension one, and the exceptional divisor is an appropriate algebraic construction whose support is the exceptional locus. The same ideas can be found in the theory of holomorphic mappings of complex manifold s. More precisely, suppose that math f X rightarrow Y math is a regular map of varieties which is birational that is, it is an isomorphism between open subsets of math X math and math Y math . A codimension 1 subvariety math Z subset X math is said to be exceptional if math f Z math has codimension at least 2 as a subvariety of math Y math . One may then define the exceptional divisor of math f math to be math sum i Z i in Div X , math where the sum is over all exceptional subvarieties of math f math , and is an element of the group of Divisor algebraic geometry Weil divisors on math X math . Consideration of exceptional divisors is crucial in birational geometry an elementary result see for instance Shafarevich, II.4.4 shows that any birational regular map that is not an isomorphism has an exceptional divisor. A particularly important example is the Blowing up blowup math sigma tilde X rightarrow X math of a subvariety math W subset X math in this case the exceptional divisor is exactly the preimage of math W math . References cite book author Shafarevich, Igor title Basic Algebraic Geometry, Vol. 1 publisher Springer Verlag year 1994 id ISBN 3 540 54812 2 Category Algebraic geometry Category Birational geometry ... more details
Unreferenced date December 2009 In mathematics , a hyperplane section of a subset X of projective space P sup n sup is the intersection set theory intersection of X with some hyperplane H &mdash in other words we look at the subset X sub H sub of those elements x of X that satisfy the single linear condition L 0 defining H as a Euclidean subspace linear subspace . Here L or H can range over the dual projective space of non zero linear form s in the homogeneous coordinates , up to scalar multiplication . From a geometrical point of view, the most interesting case is when X is an algebraic subvariety &mdash for more general cases, in mathematical analysis , some analogue of the Radon transform applies. In algebraic geometry , assuming therefore that X is V , a subvariety not lying completely in any H , the hyperplane sections are algebraic set s with irreducible component s all of dimension n &minus 1. What more can be said is addressed by a collection of results known collectively as Bertini s theorem . The topology of hyperplane sections is studied in the topic of the Lefschetz hyperplane theorem and its refinements. Because the dimension drops by one in taking hyperplane sections, the process is potentially an inductive method for understanding varieties of higher dimension. A basic tool for that is the Lefschetz pencil . DEFAULTSORT Hyperplane Section Category Algebraic geometry ... more details
In mathematics, particularly in the subfield of real analytic geometry , a subanalytic set is a set of points for example in Euclidean space defined in a way broader than for semianalytic sets roughly speaking, those satisfying conditions requiring certain real power series to be positive there . Subanalytic sets still have a reasonable local description in terms of submanifold s. Formal definitions A subset V of a given Euclidean space E is semianalytic if each point has a neighbourhood U in E such that the intersection of V and U lies in the Boolean algebra structure Boolean algebra of sets generated by subsets defined by inequalities f 0, where f is a real analytic function . There is no Tarski Seidenberg theorem for semianalytic sets, and projections of semianalytic sets are in general not semianalytic. A subset V of E is a subanalytic set if for each point there exists a relatively compact semianalytic set X in a Euclidean space of dimension at least as great as E , and a neighbourhood U in E , such that the intersection of V and U is a linear projection of X onto E from F . In particular all semianalytic sets are subanalytic. On an open dense subset, subanalytic sets are submanifolds and so they have a definite dimension at most points . Semianalytic sets are contained in a real analytic subvariety of the same dimension. However, subanalytic sets are not in general contained in any subvariety of the same dimension. On the other hand there is a theorem, to the effect that a subanalytic set A can be written as a Locally finite collection locally finite union of submanifolds. Subanalytic sets are not closed under projections, however, because a real analytic subvariety that is not relatively compact can have a projection which is not a locally finite union of submanifolds, and hence is not subanalytic. References Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets , Inst. Hautes tudes Sci. Publ. Math. 1988 , no. 67, 5&ndash 42. MR 89k 32011 ... more details
Unreferenced date January 2009 In mathematics , the nilpotent cone math mathcal N math of a finite dimensional semisimple Lie algebra math mathfrak g math is the set of elements that act nilpotently in all representation of Lie algebras representations of math mathfrak g . math In other words, math mathcal N a in mathfrak g rho a mbox is nilpotent for all representations rho mathfrak g to operatorname End V . math The nilpotent cone is an algebraic variety irreducible subvariety of math mathfrak g math considered as a math k math vector space , and is invariant mathematics invariant under the adjoint representation of a Lie algebra adjoint action of math mathfrak g math on itself. Example The nilpotent cone of math operatorname sl 2 math , the Lie algebra of 2× 2 matrix mathematics matrices with vanishing trace linear algebra trace , is the variety of all 2× 2 traceless matrices with rank linear algebra rank less than or equal to math 1. math planetmath id 4748 title Nilpotent cone Category Lie algebras algebra stub ... more details
In algebraic geometry , a complex manifold is called Fujiki class C if it is bimeromorphic to a compact K hler manifold . This notion was defined by Akira Fujiki . ref A. Fujiki, On Automorphism Groups of Compact K hler Manifolds, Inv. Math. 44 1978 225 258. MathSciNet id 481142 ref Properties Let M be a compact manifold of Fujiki class C, and math X subset M math its complex subvariety. Then X is also in Fujiki class C ref A. Fujiki, Closedness of the Douady spaces of compact Kahler spaces , Publ. Res. Inst. Math. Sci. 14 1978 79 , no. 1, 1 52. MathSciNet id 486648 ref , Lemma 4.6 . Moreover, the Douady space of X that is, the moduli space moduli of deformations of a subvariety math X subset M math , M fixed is compact and in Fujiki class C. ref A. Fujiki, On the Douady space of a compact complex space in the category C. Nagoya Math. J. 85 1982 , 189 211. MathSciNet id 86j 32048 ref Conjectures J. P. Demailly and M. Paun have shown that a manifold is in Fujiki class C if and only if it supports a K hler current . ref Demailly, Jean Pierre Paun, Mihai http arxiv.org abs math.AG 0105176 Numerical characterization of the Kahler cone of a compact Kahler manifold , Ann. of Math. 2 159 2004 , no. 3, 1247 1274. MathSciNet id 2005i 32020 ref They also conjectured that a manifold M is in Fujiki class C if it admits a nef current which is big , that is, satisfies math int M omega dim Bbb C M 0. math For a cohomology class math omega in H 2 M math which is rational, this statement is known by Grauert Riemenschneider conjecture , a holomorphic line bundle L with first Chern class math c 1 L omega math Numerically effective nef and big has maximal Kodaira dimension , hence the corresponding rational map to math Bbb P H 0 L N math is generically finite onto its image, which is algebraic, and therefore K hler. Fujiki ref A. Fujiki, On a Compact Complex Manifold in C without Holomorphic 2 Forms, Publ. RIMS 19 1983 . MathSciNet id 84m 32037 ref and Ueno ref K. Ueno, ed., Open Probl ... more details
In mathematics , a semialgebraic set is a subset S of R sup n sup for some real closed field R for example R could be the field of real numbers defined by a finite sequence of polynomial equations of the form math P x 1,...,x n 0 math and inequalities of the form math Q x 1,...,x n 0 math , or any finite union of such sets. A semialgebraic function is a function with semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry which is the appropriate framework for algebraic geometry over the real numbers. Properties Similarly to algebraic subvariety algebraic subvarieties , finite unions and intersections of semialgebraic sets are still semialgebraic sets. Furthermore, unlike subvarieties, the complement of a semialgebraic set is again semialgebraic. Finally, and most importantly, the Tarski Seidenberg theorem says that they are also closed under the projection operation in other words a semialgebraic set projected onto a linear subspace yields another such as case of elimination of quantifiers . These properties together mean that semialgebraic sets form an o minimal structure on R . A semialgebraic set or function is said to be defined over a subring A of R if there is some description as in the definition, where the polynomials can be chosen to have coefficients in A . On a dense open subset of the semialgebraic set S , it is locally a submanifold . One can define the dimension of S to be the largest dimension at points at which it is a submanifold. It is not hard to see that a semialgebraic set lies inside an algebraic subvariety of the same dimension. See also Existential theory of the reals References citation first1 J. last1 Bochnak first2 M. last2 Coste first3 M. F. last3 Roy title Real algebraic geometry publisher Springer Verlag location Berlin year 1998 . citation first1 Edward last1 Bierstone first2 Pierre D. last2 Milman title Semianalytic and subanalytic sets journal Inst. Hautes tudes Sci. Publ. Math. year 1988 volume 6 ... more details
sup 1 sup × X , two of whose fibers are Y and Z . In more classical language, we want a subvariety ... the Chow ring, we are interested in the co dimension of the subvariety that is, the difference ... an interpretation in terms of the degree of a subvariety. For example, the Chow ring of projective ... . Furthermore, any subvariety Y of degree d and codimension k is rationally equivalent to math ... a subvariety of X and Y&prime a subvariety of X&prime , math f Y cdot f Y f Y cdot Y math Cohomological ... math Y math first to the homology class determined by the closed subvariety Y , and then to its ... subvariety Y of codimension k and degree d . If k 0 then Y is necessarily equal to P sup n sup itself ... scheme mathematics scheme theory, namely, that a subvariety Y defined by a sheaf mathematics sheaf ... more details
In commutative algebra , the height of a prime ideal math mathfrak p math in a ring mathematics ring math R math is the number of strict inclusions in the longest chain of prime ideal s contained in math mathfrak p math . ref Matsumura, Hideyuki Commutative Ring Theory , page 30 31, 1989 ref Then the height of an ideal I is the infimum of the heights of all prime ideals containing I . In the language of algebraic geometry , this is the codimension of the subvariety of Spec R corresponding to I . ref Matsumura, Hideyuki Commutative Ring Theory , page 30 31, 1989 ref It is not true that every maximal chain of prime ideals with common endpoints has the same length the first counterexample was found by Masayoshi Nagata . The existence of such an ideal is usually considered pathological and is ruled out by an assumption that the ring is catenary ring theory catenary . Many conditions on rings impose conditions on the heights of certain ideals or on all ideals of certain heights. Some notable conditions are A ring is catenary ring theory catenary if and only if for every two prime ideals math mathfrak p math math mathfrak q math , every saturated chain of strict inclusions math mathfrak p mathfrak p 1 subsetneq mathfrak p 2 cdots subsetneq mathfrak p h mathfrak q math has the same length math h math . A ring is universally catenary if and only if any finitely generated algebra over it is catenary. A local ring is Cohen Macaulay ring Cohen Macaulay if and only if for any ideal I the height and depth ring theory depth of I with respect to I are equal. A Noetherian integral domain is a unique factorization domain if and only if every height 1 prime ideal is principal. ref Hartshorne,Robin Algebraic Geometry , page 7,1977 ref In a Noetherian ring , Krull s principal ideal theorem Krull s height theorem says that the height of an ideal generated by n elements is no greater than n . reflist DEFAULTSORT Height Ring Theory Category Ring theory nl Hoogte ringtheorie pt Altura teor ... more details
Unreferenced stub auto yes date December 2009 This article is about form in botany. For the use in zoology, see Form zoology . In botanical nomenclature , a form forma , plural formae is one of the secondary taxonomic rank s, below that of Variety biology variety , which in turn is below that of species it is an infraspecific taxon. If more than three ranks are listed in describing a taxon, the classification is being specified, but only three parts make up the name of the taxon a genus name, a specific epithet , and an infraspecific epithet . The abbreviation f. or the full forma should be put before the infraspecific epithet to indicate the rank. For example Acanthocalycium spiniflorum f. klimpelianum Acanthocalycium spiniflorum f. klimpelianum or Acanthocalycium spiniflorum forma klimpelianum Weidlich & Werderm. Donald Crataegus aestivalis Walter Torr. & A.Gray var. cerasoides Sarg. f. luculenta Sarg. is a classification of a plant whose name is Crataegus aestivalis Walter Torr. & A.Gray f. luculenta Sarg. A form usually designates a group with a noticeable but minor deviation. For instance, white flowered forms of species that usually have coloured flowers can be named a f. alba . Formae apomicticae are sometimes named among plants that reproduce asexually, by apomixis . Some botanists believe that there is no need to name forms, since there are theoretically countless numbers of forms based on a single gene difference. See also Trinomial nomenclature Variety botany Subvariety Variety plant Cultivar Hybrid biology Race biology Taxonomic ranks DEFAULTSORT Form Botany Category Botanical nomenclature rank28 Botany stub es Forma bot nica fr Forme botanique mt Forma ja tr Form botanik ... more details
one suitable set of identities math x yz xy z math math 1 x x 1 x math math x x 1 x 1 x 1. math A subvariety ... operation is omitted , the class of groups does not form a subvariety of the variety of semigroups because the signatures are different. On the other hand the class of abelian group s is a subvariety ... , a subclass U of V that is itself a variety is a subvariety of V implies that U is a full subcategory ... more details
about the general Pl cker embedding the classical case of 2 planes in 4 space Pl cker coordinates In mathematics, the Pl cker embedding describes a method to realize the Grassmannian of all r dimensional subspaces of a vector space V as a subvariety of the projective space of the r th exterior power of that vector space, P sup r sup V . The Pl cker embedding was first defined, in the case r 2, n 4, in Pl cker coordinates coordinates by Julius Pl cker as a way of describing the lines in three dimensional space which, as projective line s in real projective space , correspond to two dimensional subspaces of a four dimensional vector space . This was generalized by Hermann Grassmann to arbitrary r and n using a generalization of Pl cker s coordinates, sometimes called Grassmann coordinates . Definition The Pl cker embedding over the field K is the map defined by math begin align iota colon mathbf Gr r, K n & rightarrow mathbf P wedge r K n operatorname span v 1, ldots, v r & mapsto K v 1 wedge cdots wedge v r end align math where Gr r , K sup n sup is the Grassmannian , i.e., the space of all r dimensional subspaces of the n dimensional vector space , K sup n sup . This is an isomorphism from the Grassmannian to the image of , which is a projective variety. This variety can be completely characterized as an intersection of quadrics, each coming from a relation on the Pl cker or Grassmann coordinates that derives from linear algebra. References Citation last1 Griffiths first1 Phillip author1 link Phillip Griffiths last2 Harris first2 Joseph author2 link Joe Harris mathematician title Principles of algebraic geometry publisher John Wiley & Sons location New York series Wiley Classics Library isbn 978 0 471 05059 9 id MathSciNet id 1288523 year 1994 DEFAULTSORT Plucker Embedding Category Algebraic geometry Category Differential geometry ... more details
Baghdad Arabic or the Baghdadi Arabic is the Arabic Varieties of Arabic variety spoken in Baghdad , the capital of Iraq . During the last century, Baghdad Arabic has become the lingua franca of Iraq, and the language of commerce and education. It is a subvariety of Iraqi Arabic . An interesting sociolinguistic feature of Baghdad is the existence of three distinct dialects Muslim, Baghdad Arabic Jewish Jewish and Christian Baghdadi Arabic. Muslim Baghdadi belongs to a group called gilit dialects, while Jewish Baghdadi as well as Christian Baghdadi belongs to qeltu dialects. Baghdadi gilit Arabic, which is considered the standard Baghdadi Arabic, shares many features with Gulf Arabic and with varieties spoken in some parts of eastern Syria . Gilit Arabic is of Bedouin provenance, unlike Christian and Jewish Baghdadi, which is believed to be descendant of Medieval Iraqi Arabic . Until the 1950s Baghdad Arabic contained a large inventory of borrowings from English language English , Turkish language Turkish , Persian language Persian or Kurdish language . During the first decades of the 20th century, when the population of Baghdad was less than a million, some inner city quarters had their own distinctive speech characteristics, maintained for generations. From about the 1960s, with the population movement within the city, and the influx of large numbers of people hailing mainly from the south, Baghdad Arabic has become more standardized, and has come to incorporate some rural and Bedouin features. Distinct features of Muslim Baghdadi Arabic is the use of ani as opposed to the fusha ana meaning I am . Also, they add ich when directing females ani gilitlich meaning I told you whereas maslawi s would say ana qiltolki . See also Baghdad Arabic Jewish Maslawi References Kees Versteegh, et al. Encyclopedia of Arabic Language and Linguistics , BRILL, 2006. Varieties of Arabic Category Arabic languages Category Fertile Crescent Category Languages of Iraq ... more details
In mathematics, the Riemann Hilbert correspondence is a generalization of Hilbert s twenty first problem to higher dimensions. The original setting was for Riemann surface s, where it was about the existence of regular differential equation s with prescribed monodromy groups. In higher dimensions, Riemann surfaces are replaced by complex manifold s of dimension 1, and there is a correspondence between certain systems of partial differential equation s linear and having very special properties for their solutions and possible monodromies of their solutions. Such a result was proved independently by Masaki Kashiwara 1980 and Zoghman Mebkhout 1980 . Statement Suppose that X is a complex variety. Riemann Hilbert correspondence general form there is a functor DR called the de Rham functor, that is an equivalence from the category of holonomic D module s on X with regular singularities to the category of perverse sheaves on X . By considering the irreducible elements of each category, this gives a 1 1 correspondence between isomorphism classes of irreducible holonomic D module s on X with regular singularities, and intersection cohomology complexes of irreducible closed subvarieties of X with coefficients in irreducible local system s. A D module is something like a system of differential equations on X , and a local system on a subvariety is something like a description of possible monodromies, so this correspondence can be thought of as describing certain systems of differential equations in terms of the monodromies of their solutions. References Citation last1 Borel first1 Armand author1 link Armand Borel title Algebraic D Modules publisher Academic Press location Boston, MA series Perspectives in Mathematics isbn 978 0 12 117740 9 year 1987 volume 2 Citation last1 Deligne first1 Pierre author1 link Pierre Deligne title quations diff rentielles points singuliers r guliers series Springer Lecture notes in Mathematics oclc 169357 year 1970 volume 163 M. Kashiwara, Fai ... more details
In algebraic geometry normal crossings is the property of intersecting geometric objects to do it in a transversal way. Normal crossing divisors In algebraic geometry , normal crossing divisors are a class of Divisor algebraic geometry divisors which generalize the smooth divisors. Intuitively they cross only in a transversal way. Let A be an algebraic variety , and math Z cup i Z i math a Divisor algebraic geometry reduced Cartier divisor , with math Z i math its irreducible components. Then Z is called a smooth normal crossing divisor if either i A is a algebraic curve curve , or ii all math Z i math are smooth, and for each component math Z k math , math Z Z k Z k math is a smooth normal crossing divisor. Equivalently, one says that a reduced divisor has normal crossings if each point tale topology tale locally looks like the intersection of coordinate hyperplanes. Normal crossings singularity In algebraic geometry a normal crossings singularity is a point in an algebraic variety that is locally isomorphic to a normal crossings divisor. Simple normal crossings singularity In algebraic geometry a simple normal crossings singularity is a point in an algebraic variety , the latter having smooth irreducible component irreducible components , that is locally isomorphic to a normal crossings divisor. Examples The normal crossing points in the algebraic variety called the Whitney umbrella are not simple normal crossings singularities. The origin in the algebraic variety defined by math xy 0 math is a simple normal crossings singularity. The variety itself, seen as a subvariety of the two dimensional Cartesian coordinate system affine plane is an example of a normal crossings divisor. References Robert Lazarsfeld, Positivity in algebraic geometry , Springer Verlag, Berlin, 1994. Category Algebraic geometry Category Geometry of divisors ... more details
have the same dimension , we can define for any subvariety Y nowiki nowiki X nowiki nowiki ... is proper morphism proper , for Y a subvariety of X the pushforward is defined to be math f Y n f ... more details
wiktionary variety Variety may refer to Variety show , a form of theatrical and television entertainment. Variety botany , a formal rank in botanical taxonomic nomenclature, below that of species and above subvariety and form, for a usually naturally occurring variant of a plant species Variety , an informal and incorrect and ambiguous rather than taxonomic term for cultivar a cultivated variant of a plant species in horticulture, especially viticulture and rice production Plant variety law , a legal rather than taxonomic term for a cultivar or hybrid protected by patent law Variety cybernetics , the number of possible states of a system or of an element of the system Variety linguistics , a concept that includes, for instance, dialects, standard language and jargon Variety magazine Variety magazine , an entertainment industry newspaper Variety mineralogy , a mineral subform Variety numismatics , a term in coin collecting Variety philately , a term in stamp collecting Variety US radio , a format of radio programming Wine variety, more commonly varietal , one of a number of types of wine made primarily from a single named List of grape varieties grape cultivar Variety, the Children s Charity , an international charity that raises funds to help children with special needs Variety racehorse , a competitor who failed to complete the 1848 Grand National Mathematics Abelian variety , a complex torus that can be embedded into projective space Abstract variety , an intrinsically defined variety Algebraic variety , the basic object of study in algebraic geometry Algebraic variety Affine variety , a subset of algebraic varieties Algebraic variety Projective variety , a subset of algebraic varieties Quasiprojective variety , a subset of algebraic varieties which includes projective and affine varieties Variety universal algebra , classes of algebraic structures defined by equations in universal algebra Film Variet , a 1925 silent film Variety 1935 film Variety 1935 film , a 1 ... more details
Unreferenced date December 2009 About the naming of plants and fungi the equivalent naming of animals Trinomen In botanical nomenclature, the ICBN prescribes a three part name ternary name for any taxon below the rank of species. The ranks below that of species explicitly allowed in the ICBN are subspecies subspecies recommended abbreviation subsp., but ssp. is also in use varietas variety botany variety recommended abbreviation var. subvarietas subvariety recommended abbreviation subvar. forma form botany form recommended abbreviation f. subforma subforma recommended abbreviation subf. Such a taxon is called an infraspecific taxon. Its name consists of three parts a genus name, a specific epithet and an infraspecific epithet. A connecting term should be placed before the infraspecific epithet to indicate the rank. It is customary to italicize all three parts of a ternary name. For example Acanthocalycium klimpelianum var. macranthum Astrophytum myriostigma subvar. glabrum Backeb. When indicating authors, it is possible to indicate either only the final epithet s author, or both the specific and subspecific authors after their respective epithets. Homonym biology Homonymy is not allowed between subspecific epithet two forms may not have the same name even if they belong to different varieties or subspecies, and two ranks may not have the same name unless they also have the same holotype . clarify date May 2010 reason Clarify this gobbledygook explain this in plain English and provided examples of good and bad usage in both of these cases. Examples Adenia aculeata subsp. inermis de Wilde Identifying de Wilde as the author who published this name. Note that here it was decided not to indicate authority for the species Pinus nigra var. pallasiana Lambert Asch. & Graebn. Here, Lambert published the epithet in a name at the rank of species Pinus pallasiana and the taxon was subsequently reduced to a variety of Pinus nigra subsp. nigra . Pinus nigra J.F.Arnold subsp. salz ... more details
In mathematics , the Prym variety named for Friedrich Prym construction is a method in algebraic geometry of making an abelian variety from a morphism of algebraic curve s. In its original form, it was applied to an unramified Double cover topology double covering of a Riemann surface , and was used by W. Schottky and H. W. E. Jung in relation with the Schottky problem , as it now called, of characterising Jacobian varieties among abelian varieties. It is said to have appeared first in the late work of Riemann, and was extensively studied by Wirtinger in 1895, including degenerate cases. Given a non constant morphism &phi C sub 1 sub &rarr C sub 2 sub of algebraic curves, write J sub i sub for the Jacobian variety of C sub i sub . Then from construct the corresponding morphism &psi J sub 1 sub &rarr J sub 2 sub , which can be defined on a divisor class D of degree zero by applying to each point of the divisor. This is a well defined morphism, often called the norm homomorphism . Then the Prym variety of is the kernel algebra kernel of . To qualify that somewhat, to get an abelian variety , the connected component of the identity of the reduced scheme underlying the kernel scheme theory kernel may be intended. Or in other words take the largest abelian subvariety of J sub 1 sub , on which is trivial. The theory of Prym varieties was dormant for a long time, until revived by David Mumford around 1970. It now plays a substantial role in some contemporary theories, for example of the Kadomtsev Petviashvili equation . One advantage of the method is that it allows one to apply the theory of curves to the study of a wider class of abelian varieties than Jacobians. E.g. principally polarized abelian varieties of dimension 3 are not generally Jacobians, but all p.p.a.v. s of dimension 5 or less are Prym varieties. It is for this reason that p.p.a.v. s are fairly well understood up to dimension 5. References Cite book first1 Christina last1 Birkenhake first2 Herbert ... more details
unsourced date September 2010 In mathematics , in the field of algebraic geometry , the idea of abstract variety is to define a concept of algebraic variety in an intrinsic way. This followed the trend in the definition of manifold independent of any ambient space Hassler Whitney , in the 1930s by some years, the first notions being those of Oscar Zariski and Andr Weil in the 1940s. It was Weil, in his foundational work, who gave a first acceptable definition of algebraic variety that stood outside projective space . The simplest notion of algebraic variety is affine algebraic variety. If k is a given algebraically closed field, then A sup n sup k is the n fold Cartesian product of k with itself. Given an ideal I in the ring k x sub 1 sub ,...,x sub n sub of polynomials in n variables over k, the zero set V I is the affine variety defined by the ideal. Unfortunately, affine varieties lack a fundamental property known as completeness. To address this deficiency, affine varieties can be completed , by embedding them in projective space. Formally, a new variable x sub 0 sub is introduced, and the polynomials are replaced by homogeneous polynomials. Choosing an index i to omit from the defining polynomials provide an affine subspace of P sup n sup , and an open affine subvariety of the projective variety. The problem with this approach is that the mechanics of working with projective space and homogeneous coordinates is not terribly geometrical, and is also somewhat arbitrary. Taking a step back, we see that if V is projective variety, the set of affine varieties we have defined is an open cover of V. Moreover, if U sub sub is an element of the open cover, there is an associated affine coordinate ring O U sub sub , and the assignment of coordinate rings to these sets forms a presheaf on V which will be known as the structure sheaf . An abstract variety V,O is a topological space V with an associated sheaf O of commutative rings that has the additional property that ... more details
main Linear system of divisors In mathematics , specifically algebraic geometry , the base locus of a linear system of divisors on a Algebraic variety variety refers to the subvariety of points common to all divisors in the linear system. Geometrically, this corresponds to the common intersection of the varieties. Definition More precisely, suppose that math D math is a linear system of divisors on some variety math X math . Consider the intersection math textrm Bl D bigcap D text eff in D textrm Supp D text eff math where math textrm Supp math denotes the support of a divisor, and the intersection is taken over all effective divisors math D text eff math in the linear system. This is the base locus of math D math as a set, at least there may be more subtle Scheme mathematics scheme theoretic considerations as to what the Ringed space structure sheaf of math textrm Bl math should be . One application of the notion of base locus is to Numerically effective nefness of a Cartier divisor class i.e. complete linear system . Suppose math D math is such a class on a variety math X math , and math C math an irreducible curve on math X math . If math C math is not contained in the base locus of math D math , then there exists some divisor math tilde D math in the class which does not contain math C math , and so intersects it properly. Basic facts from intersection theory then tell us that we must have math D cdot C geq 0 math . The conclusion is that to check nefness of a divisor class, it suffices to compute the intersection number with curves contained in the base locus of the class. So, roughly speaking, the smaller the base locus, the more likely it is that the class is nef. In the modern formulation of algebraic geometry, a linear system math D math of Cartier divisors on a variety math X math is viewed as a line bundle math L D math on math X math . From this viewpoint, the base locus math textrm Bl D math is the set of common zeroes of all sections of math L D math . ... more details
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