For mathematical use algebraic variety universal algebra Unreferenced date December 2009 A subvariety Latin subvarietas in botanical nomenclature is a taxonomic rank . They are rarely used to classify organisms. Plant taxonomy Subvariety is ranked below that of Variety biology variety varietas above that of Form botany form forma . Subvariety is an infraspecific taxon . Name Its name consists of three parts a genus name genera a specific epithet species an infraspecific epithet subvariety To indicate the subvariety rank, the abbreviation subvar. is put before the infraspecific epithet. Taxonomic ranks Category Botanical nomenclature Species Category Plant taxonomy 1rank27 Category Biology terminology Botany stub ... 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
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, 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 F 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 3201 ... 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
Mendocino may mean Mendocino County, California , USA Mendocino, California , a town in Mendocino County Cape Mendocino in Humboldt County, California Mendocino Brewing Company , a brewery and brewpub located in the county of the same name Mendocino National Forest , a national forest Mendocino Fracture Zone , a seismic fracture zone off the coast of Cape Mendocino Mendocino Range , a mountain range in the Pacific Coast Range Mendocino album Mendocino album , 1969 album by Sir Douglas Quintet Mendocino song , by Sir Douglas Quintet Mendocino, a code name for the second generation Celeron Mendocino Celeron processor by Intel Wine regions Several wine regions located in Mendocino county. Mendocino County wine , general overview of Mendocino wine Mendocino Ridge AVA Mendocino AVA Other usages Torront s Mendocino , a subvariety of the white Argentine grape variety Torront s disambig es Mendocino fr Mendocino it Mendocino pl Mendocino ... 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 . 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 attribution 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
a subvariety V of the product family two of whose fibers are Y and Z , and all of whose fibers are subvarieties ... of constructing the Chow ring, we are interested in the co dimension of the subvariety that is, the difference ... acquiring an interpretation in terms of the degree of a subvariety. For example, the Chow ... linear functional . Furthermore, any subvariety Y of degree d and codimension k is rationally equivalent ... formula for Y a subvariety of X and Y&prime a subvariety of X&prime , math f Y cdot f Y f Y cdot Y math ... equivalence class math Y math first to the homology class determined by the closed subvariety Y , and then to its ... an arbitrary subvariety Y of codimension k and degree d . If k 0 then Y is necessarily equal to P sup ... is via scheme mathematics scheme theory, namely, that a subvariety Y defined by a sheaf mathematics ... 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 math R math 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 A ... 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 name botany 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 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 minor genetic differences. See also Trinomial nomenclature Variety botany Subvariety Variety plant Cultivar Hybrid biology Race biology Taxonomic ranks DEFAULTSORT Form Botany Category Botanical nomenclature Form Category Plant taxonomy 1rank28 Botany stub es Forma bot nica eo Formo botaniko fr Forme botanique hu Alak rendszertan mt Forma botanika nl Vorm biologie tr Form botanik ... more details
1 x 1. math A subvariety of a variety V is a subclass of V that has the same signature as V and is itself ... is omitted and or the inverse operation is omitted , the class of groups does not form a subvariety ... group s is a subvariety of the variety of groups because it consists of those groups satisfying ... theory category , a subclass U of V that is itself a variety is a subvariety of V implies that U ... more details
otheruses4 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
In mathematics , in the field of algebraic geometry , a normal scheme is a scheme mathematics scheme X for which every stalk local ring O sub X,x sub of its structure sheaf O sub X sub is an integrally closed domain integrally closed local ring that is, each stalk is an integral domain such that its integral closure in its field of fractions is equal to itself. An older and unrelated meaning of normal is that a normal variety is a subvariety of projective space such that the linear system giving the embedding is complete see rational normal surface and rational normal curve for examples. An example of a normal scheme is a regular scheme . Any reduced scheme has a normalization , whose construction we first give for irreducible reduced schemes. An irreducible and reduced scheme math X math has the property that every affine chart is a integral domain domain . Choose an affine cover corresponding to rings math A i math . Compute the integral closure of each of these in its fraction field, denote them by math overline A i math . It is not hard to see that one can construct a new scheme math overline X math by gluing together the affine schemes Spec math overline A i math . If the initial scheme is not irreducible, one can define the normalization as the disjoint union of the normalizations of the irreducible components. An alternate, equivalent, definition uses integral closures in rings of fractions where any nonzero divisor is allowed in the denominator. See also Integral closure Semi normal scheme References Hartshorne AG , p. 91 DEFAULTSORT Normal Scheme Category Scheme theory algebra stub ... more details
No footnotes date October 2011 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. cite book last Ab Haidar first Far da title Christian Arabic of Baghdad year 1991 publish ... 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
the same dimension , we can define for any subvariety Y nowiki nowiki X nowiki nowiki ... morphism proper , for Y a subvariety of X the pushforward is defined to be math f Y n f Y , math ... more details
additional points for each subvariety, called the generic point of the subvariety. Then a generic ... is true generically on the subvariety in the sense of being true on an open dense subset ... more details
In mathematics , the Jacobian variety J C of a non singular algebraic curve C of genus mathematics genus g is the moduli space of degree 0 line bundle s. It is the connected component of the identity in the Picard group of C , hence an abelian variety . Introduction The Jacobian variety is named after Carl Gustav Jacobi , who proved the complete version Abel Jacobi theorem , making the injectivity statement of Niels Abel into an isomorphism. It is a principally polarized abelian variety , of dimension g , and hence, over the complex numbers, it is a complex torus . If p is a point of C , then the curve C can be mapped to a subvariety of J with the given point p mapping to the identity of J , and C generates J as a Group mathematics group . Construction over for complex curves Over the complex numbers, the Jacobian variety can be realized as the quotient space V L , where V is the dual of the vector space of all global holomorphic differentials on C and L is the lattice of all elements of V of the form math omega mapsto int gamma omega math where is a closed path topology path in C . The Jacobian of a curve over an arbitrary field was constructed by harvtxt Weil 1948 as part of his proof of the Riemann hypothesis for curves over a finite field. The Abel Jacobi theorem states that the torus thus built is a variety, the classical Jacobian of a curve, that indeed parametrizes the degree 0 line bundles, that is, it can be identified with its Picard variety of degree 0 divisors modulo linear equivalence. Further notions Torelli s theorem states that a complex curve is determined by its Jacobian with its polarization . The Schottky problem asks which principally polarized abelian varieties are the Jacobians of curves. The Picard variety , the Albanese variety , and intermediate Jacobian s are generalizations of the Jacobian for higher dimensional varieties. For varieties of higher dimension the construction of the Jacobian variety as a quotient of the space of holomorphi ... more details
In mathematics , the theta divisor is the divisor algebraic geometry divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers and principally polarized by the zero locus of the associated Riemann theta function . It is therefore an algebraic subvariety of A of dimension dim A &minus 1. Classical theory Classical results of Bernhard Riemann describe in another way, in the case that A is the Jacobian variety J of an algebraic curve compact Riemann surface C . There is, for a choice of base point P on C , a standard mapping of C to J , by means of the interpretation of J as the linear equivalence classes of divisors on C of degree 0. That is, Q on C maps to the class of Q &minus P . Then since J is an algebraic group , C may be added to itself k times on J , giving rise to subvarieties W sub k sub . If g is the genus mathematics genus of C , Riemann proved that is a translate on J of W sub g &minus 1 sub . He also described which points on W sub g &minus 1 sub are non singular they correspond to the effective divisors D of degree g &minus 1 with no associated meromorphic functions other than constants. In more classical language, these D do not move in a linear system of divisors on C , in the sense that they do not dominate the polar divisor of a non constant function. Riemann further proved the Riemann singularity theorem , identifying the multiplicity of a point p class D on W sub g &minus 1 sub as the number of independent meromorphic functions with pole divisor dominated by D, or equivalently as h sup 0 sup O D , the number of independent global section s of the holomorphic line bundle associated to D as Cartier divisor on C . Later work The Riemann singularity theorem was extended by George Kempf in 1973, ref cite journal author G. Kempf title On the geometry of a theorem of Riemann journal Ann. Of Math. volume 98 year 1973 pages 178 185 doi 10.2307 1970910 jstor 1970910 issue 1 ref building on work of David Mumfor ... more details
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