In algebraic geometry, given a reductive algebraic group G and a Borel subgroup B, a spherical variety is a G-variety with an open dense B-orbit. It is sometimes also assumed to be normal. Examples are flag varieties, symmetric spaces and (affine or projective) toric varieties.
There is also a notion of real spherical varieties.
A projective spherical variety is a Mori dream space.[1]
Spherical embeddings are classified by so-called colored fans, a generalization of fans for toric varieties; this is known as Luna-Vust Theory.
In his seminal paper, Luna (2001) developed a framework to classify complex spherical subgroups of reductive groups; he reduced the classification of spherical subgroups to wonderful subgroups. He further worked out the case of groups of type A and conjectured that combinatorial objects consisting of "homogeneous spherical data" classify spherical subgroups. This is known as the Luna Conjecture.
This classification is now complete according to Luna's program; see contributions of Bravi, Cupit-Foutou, Losev and Pezzini.
As conjectured by Knop, every "smooth" affine spherical variety is uniquely determined by its weight monoid.
This uniqueness result was proven by Losev.
Knop (2013) has been developing a program to classify spherical varieties in arbitrary characteristic.
^Brion, Michel (2007). "The total coordinate ring of a wonderful variety". Journal of Algebra. 313 (1): 61–99. arXiv:math/0603157. doi:10.1016/j.jalgebra.2006.12.022. S2CID 15154549.
subgroup B, a sphericalvariety is a G-variety with an open dense B-orbit. It is sometimes also assumed to be normal. Examples are flag varieties, symmetric...
scientific fields. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of...
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles...
Spherical Earth or Earth's curvature refers to the approximation of the figure of the Earth as a sphere. The earliest documented mention of the concept...
The roughly spherical shape of Earth can be empirically evidenced by many different types of observation, ranging from ground level, flight, or orbit...
and time. Therefore, we can generate a great variety of solutions by translating and summing spherical waves. Let φ(ξ, η, ζ) be an arbitrary function...
Spherical astronomy, or positional astronomy, is a branch of observational astronomy used to locate astronomical objects on the celestial sphere, as seen...
Homogeneous varietySphericalvariety Ash, A.; Mumford, David; Rapoport, M.; Tai, Y. (1975), Smooth compactification of locally symmetric varieties (PDF),...
called the index of speciality, is positive. sphericalvariety A sphericalvariety is a normal G-variety (G connected reductive) with an open dense orbit...
that groceries often sell half or quarter melons. Some smaller, sphericalvarieties of watermelon—both red- and yellow-fleshed—are sometimes called "icebox...
graphics differences. Most motion picture lenses are of the sphericalvariety. Spherical lenses do not distort the image intentionally. Used alone for...
A spherical tokamak is a type of fusion power device based on the tokamak principle. It is notable for its very narrow profile, or aspect ratio. A traditional...
In an automobile, ball joints are spherical bearings that connect the control arms to the steering knuckles, and are used on virtually every automobile...
Mexican husk tomato, is a plant of the nightshade family bearing small, spherical, and green or green-purple fruit of the same name. Tomatillos originated...
independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian...
hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines...
information-theoretic quantities beyond the existing information theory. The spherical scoring rule is also a strictly proper scoring rule S ( r , i ) = r i...
Akhiezer, D. N.; Vinberg, E. B. (1999), "Weakly symmetric spaces and sphericalvarieties", Transf. Groups, 4: 3–24, doi:10.1007/BF01236659 van den Ban, E...
multiplied by the anamorphic power of the camera lenses (1× in the case of spherical lenses). Gate dimensions are the width and height of the camera gate aperture...
differential equations) in spherical coordinates. Associated Legendre polynomials play a vital role in the definition of spherical harmonics. These functions...
arithmetic of linear algebraic groups, homogeneous spaces, Shimura varieties, and sphericalvarieties. Borovoi was born in Moscow. He obtained his diploma (M.Sc...
protect from witches bringing bad luck to the hunt. There is a more sphericalvariety with a similar, but shorter side extension called a ndemba.: 88 In...
Baroque pearls are pearls with an irregular, non-spherical shape. Shapes can range from minor aberrations to distinctly ovoid, curved, pinched, or lumpy...
Although these viruses are often referred to as 'spherical', they do not show true mathematical spherical symmetry. In the early 20th century, Ernst Haeckel...
been shown that a variety which admits a surjective morphism from a Mori dream space is again a Mori dream space. Sphericalvariety Castravet, Ana-Maria...
tetrahedron or tristetrahedron. Among the varieties, there are also often concentric, fibrous or spherical aggregates of a very impressive appearance...
coloured usually either wine red (for spherical particles less than 100 nm) or blue-purple (for larger spherical particles or nanorods). Due to their optical...
course and can be accompanied by a variety of sauces and side dishes. The "heart" sweetbreads are more spherical, while the "throat" sweetbreads are...
by artificial selection for lateral meristem growth (a swollen, nearly spherical shape); its origin in nature is the same as that of cabbage, broccoli...
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