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Endraß Surface
In algebraic geometry, an Endrass surface is a nodal surface of degree 8 with 168 real nodes, found by . , it remained the record-holder for the most number of real nodes for its degree; however, the best proven upper bound, 174, does not match the lower bound given by this surface. See also *Barth surface *Sarti surface In algebraic geometry, a Sarti surface is a degree-12 nodal surface with 600 nodes, found by . The maximal possible number of nodes of a degree-12 surface is not known (as of 2015), though Yoichi Miyaoka showed that it is at most 645. Sarti has ... * Togliatti surface References {{reflist Algebraic surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endrass Surface
In algebraic geometry, an Endrass surface is a nodal surface of degree 8 with 168 real nodes, found by . , it remained the record-holder for the most number of real nodes for its degree; however, the best proven upper bound, 174, does not match the lower bound given by this surface. See also *Barth surface * Sarti surface *Togliatti surface In algebraic geometry, a Togliatti surface is a nodal surface of degree five with 31 nodes. The first examples were constructed by . proved that 31 is the maximum possible number of nodes for a surface of this degree, showing this example to b ... References {{reflist Algebraic surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nodal Surface
In algebraic geometry, a nodal surface is a surface in (usually complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...) projective space whose only singularities are nodes. A major problem about them is to find the maximum number of nodes of a nodal surface of given degree. The following table gives some known upper and lower bounds for the maximal number of nodes on a complex surface of given degree. In degree 7, 9, 11, and 13, the upper bound is given by , which is better than the one by . See also * Algebraic surface References * * * *{{citation , mr=3124329 , doi=10.1016/j.crma.2013.09.009 , last=Escudero , first=Juan GarcÃa , title=On a family of complex algebraic surfaces of degree 3''n'' , journal=C. R. Math. Acad. Sci. Paris , volume=351 , year=2013 , i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barth Surface
__NOTOC__ In algebraic geometry, a Barth surface is one of the complex nodal surfaces in 3 dimensions with large numbers of double points found by . Two examples are the Barth sextic of degree 6 with 65 double points, and the Barth decic of degree 10 with 345 double points. For degree 6 surfaces in P3, showed that 65 is the maximum number of double points possible. The Barth sextic is a counterexample to an incorrect claim by Francesco Severi in 1946 that 52 is the maximum number of double points possible. Informal accounting of the 65 ordinary double points of the Barth Sextic The Barth Sextic may be visualized in three dimensions as featuring 50 finite and 15 infinite ordinary double points (nodes). Referring to the figure, the 50 finite ordinary double points are arrayed as the vertices of 20 roughly tetrahedral shapes oriented such that the bases of these four-sided "outward pointing" shapes form the triangular faces of a regular icosidodecahedron. To these 30 icosidodeca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sarti Surface
In algebraic geometry, a Sarti surface is a degree-12 nodal surface with 600 nodes, found by . The maximal possible number of nodes of a degree-12 surface is not known (as of 2015), though Yoichi Miyaoka showed that it is at most 645. Sarti has also found sextic, octic and dodectic nodal surfaces with high numbers of nodes and high degrees of symmetry. File:Sarti sextic 48 A.png, Sextic with 48 node File:Sarti sextic 48 (Stabchen).png, Sextic with 48 node File:Sarti's Octic with 72.png, Octic with 72 nodes File:Sarti's octic with 144 nodes.png, Octic with 144 nodes File:Sarti dodectic 360.png, Dodectic surface with 360 nodes File:3D model of Sarti surface.stl, 3D model of Sarti surface See also *Nodal surface In algebraic geometry, a nodal surface is a surface in (usually complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex ... References * * * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Togliatti Surface
In algebraic geometry, a Togliatti surface is a nodal surface of degree five with 31 Node (algebraic geometry), nodes. The first examples were constructed by . proved that 31 is the maximum possible number of nodes for a surface of this degree, showing this example to be optimal. See also *Barth surface *Endrass surface *Sarti surface *List of algebraic surfaces References *. *. External links * * Algebraic surfaces Complex surfaces {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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