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Coble Variety
In mathematics, the Coble variety is the moduli space of ordered sets of 6 points in the projective plane, and can be represented as a double cover of the projective 4-space branched over the Igusa quartic. It is a 4-dimensional variety that was first studied by Arthur Coble. See also * Coble curve * Coble surface * Coble hypersurface References *{{Citation , last1=Hunt , first1=Bruce , title=The geometry of some special arithmetic quotients , publisher=Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ... , location=Berlin, New York , series=Lecture Notes in Mathematics , isbn=978-3-540-61795-2 , doi=10.1007/BFb0094399 , mr=1438547 , year=1996 , volume=1637 Algebraic varieties ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Motivation Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space correspond to solutions of geometric problems. Here different solutions are identified if they a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Igusa Quartic
In algebraic geometry, the Igusa quartic (also called the Castelnuovo–Richmond quartic ''CR''4 or the Castelnuovo–Richmond–Igusa quartic) is a quartic hypersurface in 4-dimensional projective space, studied by . It is closely related to the moduli space of genus 2 curves with level 2 structure. It is the dual of the Segre cubic In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by . Definition The Segre cubic is the set of points (''x''0:''x''1:''x''2:''x''3:''x''4:''x''5) of ''P''5 satisfyin .... It can be given as a codimension 2 variety in ''P''5 by the equations :\sum x_i=0 :\big(\sum x_i^2\big)^2 = 4 \sum x_i^4 References * * * 3-folds {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variety (universal Algebra)
In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to #Birkhoff's_theorem, Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphism, homomorphic images, subalgebras and Direct product#Direct product in universal algebra, (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a Category (mathematics), category; these are usually called ''finitary algebraic categories''. A ''covariety'' is the class of all F-coalgebra, coalgebraic structures of a given signature. Terminology A variety of algebras should not be confused with an algebraic variety, which means a set of solutions to a system of polynomial eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arthur Coble
Arthur Byron Coble (November 3, 1878 – December 8, 1966) was an American mathematician. He did research on finite geometries and the group theory related to them, Cremona transformations associated with the Galois theory of equations, and the relations between hyperelliptic theta functions, irrational binary invariants, the Weddle surface and the Kummer surface. He was President of the American Mathematical Society from 1933 to 1934. Biography Early life Arthur Coble was born on November 3, 1878, in Williamstown, Pennsylvania. His mother Emma was a schoolteacher. When Coble was born, his father Ruben was the manager of a store. Later, he became president of a bank. Coble's parents belonged to Evangelical Lutheran Church. Coble was brought up strictly as an Evangelical Lutheran; however, he rejected this Church when he reached adulthood. Coble entered Gettysburg College in 1893, and completed his A.B. in 1897. He spent a year as a public school teacher. He entered Johns Ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coble Curve
In algebraic geometry, a Coble curve is an irreducible degree-6 planar curve with 10 double points (some of them may be infinitely near points). They were studied by . See also * Coble surface References * *{{Citation , last1=Coble , first1=Arthur B. , title=Algebraic geometry and theta functions , origyear=1929 , publisher=American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ... , location=Providence, R.I. , series=American Mathematical Society Colloquium Publications , isbn=978-0-8218-1010-1 , mr=733252 , year=1982 , volume=10 Sextic curves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coble Surface
In algebraic geometry, a Coble surface was defined by to be a smooth rational projective surface with empty anti-canonical linear system , −K, and non-empty anti-bicanonical linear system , −2K, . An example of a Coble surface is the blowing up of the projective plane at the 10 nodes of a Coble curve. References *{{Citation , doi=10.1353/ajm.2001.0002 , last1=Dolgachev , first1=Igor V. , last2=Zhang , first2=De-Qi , title=Coble Rational Surfaces , jstor=25099046 , publisher=The Johns Hopkins University Press , year=2001 , journal=American Journal of Mathematics The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United S ... , issn=0002-9327 , volume=123 , issue=1 , pages=79–114, mr=1827278, arxiv=math/9909135 Algebraic surfaces Complex surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coble Hypersurface
In algebraic geometry, a Coble hypersurface is one of the hypersurfaces associated to the Jacobian variety of a curve of genus 2 or 3 by Arthur Coble. There are two similar but different types of Coble hypersurfaces. *The Kummer variety of the Jacobian of a genus 3 curve can be embedded in 7-dimensional projective space under the 2-theta map, and is then the singular locus of a 6-dimensional quartic hypersurface , called a Coble hypersurface. *Similarly the Jacobian of a genus 2 curve can be embedded in 8-dimensional projective space under the 3-theta map, and is then the singular locus of a 7-dimensional cubic hypersurface , also called a Coble hypersurface. See also *Coble curve (dimension 1) *Coble surface (dimension 2) * Coble variety (dimension 4) References * * *{{Citation , last1=Coble , first1=Arthur B. , title=Algebraic geometry and theta functions , orig-year=1929 , publisher=American Mathematical Society The American Mathematical Society (AMS) is an assoc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
